Algebra Examples

$x^{\frac{3}{7}} = - 8$

Step 1

Add $8$ to both sides of the equation.

$x^{\frac{3}{7}} + 8 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Rewrite $x^{\frac{3}{7}}$ as ${(x^{\frac{1}{7}})}^{3}$ .

${(x^{\frac{1}{7}})}^{3} + 8 = 0$

Step 2.2

Rewrite $8$ as $2^{3}$ .

${(x^{\frac{1}{7}})}^{3} + 2^{3} = 0$

Step 2.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x^{\frac{1}{7}}$ and $b = 2$ .

$(x^{\frac{1}{7}} + 2) ({(x^{\frac{1}{7}})}^{2} - x^{\frac{1}{7}} \cdot 2 + 2^{2}) = 0$

Step 2.4

Simplify.

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Step 2.4.1

Multiply the exponents in ${(x^{\frac{1}{7}})}^{2}$ .

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Step 2.4.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$(x^{\frac{1}{7}} + 2) (x^{\frac{1}{7} \cdot 2} - x^{\frac{1}{7}} \cdot 2 + 2^{2}) = 0$

Step 2.4.1.2

Combine $\frac{1}{7}$ and $2$ .

$(x^{\frac{1}{7}} + 2) (x^{\frac{2}{7}} - x^{\frac{1}{7}} \cdot 2 + 2^{2}) = 0$

Step 2.4.2

Multiply $2$ by $- 1$ .

$(x^{\frac{1}{7}} + 2) (x^{\frac{2}{7}} - 2 x^{\frac{1}{7}} + 2^{2}) = 0$

Step 2.4.3

Raise $2$ to the power of $2$ .

$(x^{\frac{1}{7}} + 2) (x^{\frac{2}{7}} - 2 x^{\frac{1}{7}} + 4) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{\frac{1}{7}} + 2 = 0$

$x^{\frac{2}{7}} - 2 x^{\frac{1}{7}} + 4 = 0$

Step 4

Set $x^{\frac{1}{7}} + 2$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{\frac{1}{7}} + 2$ equal to $0$ .

$x^{\frac{1}{7}} + 2 = 0$

Step 4.2

Solve $x^{\frac{1}{7}} + 2 = 0$ for $x$ .

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Step 4.2.1

Subtract $2$ from both sides of the equation.

$x^{\frac{1}{7}} = - 2$

Step 4.2.2

Raise each side of the equation to the power of $7$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{7}})}^{7} = {(- 2)}^{7}$

Step 4.2.3

Simplify the exponent.

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Step 4.2.3.1

Simplify the left side.

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Step 4.2.3.1.1

Simplify ${(x^{\frac{1}{7}})}^{7}$ .

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Step 4.2.3.1.1.1

Multiply the exponents in ${(x^{\frac{1}{7}})}^{7}$ .

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Step 4.2.3.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{7} \cdot 7} = {(- 2)}^{7}$

Step 4.2.3.1.1.1.2

Cancel the common factor of $7$ .

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Step 4.2.3.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{7} \cdot 7} = {(- 2)}^{7}$

Step 4.2.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = {(- 2)}^{7}$

Step 4.2.3.1.1.2

Simplify.

$x = {(- 2)}^{7}$

Step 4.2.3.2

Simplify the right side.

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Step 4.2.3.2.1

Raise $- 2$ to the power of $7$ .

$x = - 128$

Step 5

Set $x^{\frac{2}{7}} - 2 x^{\frac{1}{7}} + 4$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{\frac{2}{7}} - 2 x^{\frac{1}{7}} + 4$ equal to $0$ .

$x^{\frac{2}{7}} - 2 x^{\frac{1}{7}} + 4 = 0$

Step 5.2

Solve $x^{\frac{2}{7}} - 2 x^{\frac{1}{7}} + 4 = 0$ for $x$ .

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Step 5.2.1

Find a common factor $x^{\frac{1}{7}}$ that is present in each term.

${(x^{\frac{1}{7}})}^{2} - 2 x^{\frac{1}{7}} + 4$

Step 5.2.2

Substitute $u$ for $x^{\frac{1}{7}}$ .

${(u)}^{2} - 2 u + 4 = 0$

Step 5.2.3

Solve for $u$ .

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Step 5.2.3.1

Remove parentheses.

$u^{2} - 2 u + 4 = 0$

Step 5.2.3.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5.2.3.3

Substitute the values $a = 1$ , $b = - 2$ , and $c = 4$ into the quadratic formula and solve for $u$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1}$

Step 5.2.3.4

Simplify.

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Step 5.2.3.4.1

Simplify the numerator.

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Step 5.2.3.4.1.1

Raise $- 2$ to the power of $2$ .

$u = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 5.2.3.4.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 5.2.3.4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 5.2.3.4.1.2.2

Multiply $- 4$ by $4$ .

$u = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 5.2.3.4.1.3

Subtract $16$ from $4$ .

$u = \frac{2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 5.2.3.4.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$u = \frac{2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 5.2.3.4.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$u = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 5.2.3.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 5.2.3.4.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.2.3.4.1.7.1

Factor $4$ out of $12$ .

$u = \frac{2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 5.2.3.4.1.7.2

Rewrite $4$ as $2^{2}$ .

$u = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 5.2.3.4.1.8

Pull terms out from under the radical.

$u = \frac{2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 5.2.3.4.1.9

Move $2$ to the left of $i$ .

$u = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 5.2.3.4.2

Multiply $2$ by $1$ .

$u = \frac{2 \pm 2 i \sqrt{3}}{2}$

Step 5.2.3.4.3

Simplify $\frac{2 \pm 2 i \sqrt{3}}{2}$ .

$u = 1 \pm i \sqrt{3}$

Step 5.2.3.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.2.3.5.1

Simplify the numerator.

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Step 5.2.3.5.1.1

Raise $- 2$ to the power of $2$ .

$u = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 5.2.3.5.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 5.2.3.5.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 5.2.3.5.1.2.2

Multiply $- 4$ by $4$ .

$u = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 5.2.3.5.1.3

Subtract $16$ from $4$ .

$u = \frac{2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 5.2.3.5.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$u = \frac{2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 5.2.3.5.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$u = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 5.2.3.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 5.2.3.5.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.2.3.5.1.7.1

Factor $4$ out of $12$ .

$u = \frac{2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 5.2.3.5.1.7.2

Rewrite $4$ as $2^{2}$ .

$u = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 5.2.3.5.1.8

Pull terms out from under the radical.

$u = \frac{2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 5.2.3.5.1.9

Move $2$ to the left of $i$ .

$u = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 5.2.3.5.2

Multiply $2$ by $1$ .

$u = \frac{2 \pm 2 i \sqrt{3}}{2}$

Step 5.2.3.5.3

Simplify $\frac{2 \pm 2 i \sqrt{3}}{2}$ .

$u = 1 \pm i \sqrt{3}$

Step 5.2.3.5.4

Change the $\pm$ to $+$ .

$u = 1 + i \sqrt{3}$

Step 5.2.3.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.2.3.6.1

Simplify the numerator.

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Step 5.2.3.6.1.1

Raise $- 2$ to the power of $2$ .

$u = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 5.2.3.6.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 5.2.3.6.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 5.2.3.6.1.2.2

Multiply $- 4$ by $4$ .

$u = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 5.2.3.6.1.3

Subtract $16$ from $4$ .

$u = \frac{2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 5.2.3.6.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$u = \frac{2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 5.2.3.6.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$u = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 5.2.3.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 5.2.3.6.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.2.3.6.1.7.1

Factor $4$ out of $12$ .

$u = \frac{2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 5.2.3.6.1.7.2

Rewrite $4$ as $2^{2}$ .

$u = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 5.2.3.6.1.8

Pull terms out from under the radical.

$u = \frac{2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 5.2.3.6.1.9

Move $2$ to the left of $i$ .

$u = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 5.2.3.6.2

Multiply $2$ by $1$ .

$u = \frac{2 \pm 2 i \sqrt{3}}{2}$

Step 5.2.3.6.3

Simplify $\frac{2 \pm 2 i \sqrt{3}}{2}$ .

$u = 1 \pm i \sqrt{3}$

Step 5.2.3.6.4

Change the $\pm$ to $-$ .

$u = 1 - i \sqrt{3}$

Step 5.2.3.7

The final answer is the combination of both solutions.

$u = 1 + i \sqrt{3}, 1 - i \sqrt{3}$

Step 5.2.4

Substitute $x$ for $u$ .

$x^{\frac{1}{7}} = 1 + i \sqrt{3}, 1 - i \sqrt{3}$

Step 5.2.5

Solve for $x^{\frac{1}{7}} = 1 + i \sqrt{3}$ for $x$ .

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Step 5.2.5.1

Raise each side of the equation to the power of $7$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{7}})}^{7} = {(1 + i \sqrt{3})}^{7}$

Step 5.2.5.2

Simplify the exponent.

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Step 5.2.5.2.1

Simplify the left side.

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Step 5.2.5.2.1.1

Simplify ${(x^{\frac{1}{7}})}^{7}$ .

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Step 5.2.5.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{7}})}^{7}$ .

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Step 5.2.5.2.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{7} \cdot 7} = {(1 + i \sqrt{3})}^{7}$

Step 5.2.5.2.1.1.1.2

Cancel the common factor of $7$ .

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Step 5.2.5.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{7} \cdot 7} = {(1 + i \sqrt{3})}^{7}$

Step 5.2.5.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = {(1 + i \sqrt{3})}^{7}$

Step 5.2.5.2.1.1.2

Simplify.

$x = {(1 + i \sqrt{3})}^{7}$

Step 5.2.5.2.2

Simplify the right side.

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Step 5.2.5.2.2.1

Simplify ${(1 + i \sqrt{3})}^{7}$ .

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Step 5.2.5.2.2.1.1

Use the Binomial Theorem.

$x = 1^{7} + 7 \cdot 1^{6} (i \sqrt{3}) + 21 \cdot 1^{5} {(i \sqrt{3})}^{2} + 35 \cdot 1^{4} {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2

Simplify terms.

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Step 5.2.5.2.2.1.2.1

Simplify each term.

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Step 5.2.5.2.2.1.2.1.1

One to any power is one.

$x = 1 + 7 \cdot 1^{6} (i \sqrt{3}) + 21 \cdot 1^{5} {(i \sqrt{3})}^{2} + 35 \cdot 1^{4} {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.2

One to any power is one.

$x = 1 + 7 \cdot 1 (i \sqrt{3}) + 21 \cdot 1^{5} {(i \sqrt{3})}^{2} + 35 \cdot 1^{4} {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.3

Multiply $7$ by $1$ .

$x = 1 + 7 (i \sqrt{3}) + 21 \cdot 1^{5} {(i \sqrt{3})}^{2} + 35 \cdot 1^{4} {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.4

One to any power is one.

$x = 1 + 7 i \sqrt{3} + 21 \cdot 1 {(i \sqrt{3})}^{2} + 35 \cdot 1^{4} {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.5

Multiply $21$ by $1$ .

$x = 1 + 7 i \sqrt{3} + 21 {(i \sqrt{3})}^{2} + 35 \cdot 1^{4} {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.6

Apply the product rule to $i \sqrt{3}$ .

$x = 1 + 7 i \sqrt{3} + 21 (i^{2} {\sqrt{3}}^{2}) + 35 \cdot 1^{4} {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.7

Rewrite $i^{2}$ as $- 1$ .

$x = 1 + 7 i \sqrt{3} + 21 (- 1 {\sqrt{3}}^{2}) + 35 \cdot 1^{4} {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.8

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.2.5.2.2.1.2.1.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = 1 + 7 i \sqrt{3} + 21 (- 1 {(3^{\frac{1}{2}})}^{2}) + 35 \cdot 1^{4} {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.8.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = 1 + 7 i \sqrt{3} + 21 (- 1 \cdot 3^{\frac{1}{2} \cdot 2}) + 35 \cdot 1^{4} {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.8.3

Combine $\frac{1}{2}$ and $2$ .

$x = 1 + 7 i \sqrt{3} + 21 (- 1 \cdot 3^{\frac{2}{2}}) + 35 \cdot 1^{4} {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.8.4

Cancel the common factor of $2$ .

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Step 5.2.5.2.2.1.2.1.8.4.1

Cancel the common factor.

Step 5.2.5.2.2.1.2.1.8.4.2

Rewrite the expression.

$x = 1 + 7 i \sqrt{3} + 21 (- 1 \cdot 3^{1}) + 35 \cdot 1^{4} {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.8.5

Evaluate the exponent.

$x = 1 + 7 i \sqrt{3} + 21 (- 1 \cdot 3) + 35 \cdot 1^{4} {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.9

Multiply $21 (- 1 \cdot 3)$ .

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Step 5.2.5.2.2.1.2.1.9.1

Multiply $- 1$ by $3$ .

$x = 1 + 7 i \sqrt{3} + 21 \cdot - 3 + 35 \cdot 1^{4} {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.9.2

Multiply $21$ by $- 3$ .

$x = 1 + 7 i \sqrt{3} - 63 + 35 \cdot 1^{4} {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.10

One to any power is one.

$x = 1 + 7 i \sqrt{3} - 63 + 35 \cdot 1 {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.11

Multiply $35$ by $1$ .

$x = 1 + 7 i \sqrt{3} - 63 + 35 {(i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.12

Apply the product rule to $i \sqrt{3}$ .

$x = 1 + 7 i \sqrt{3} - 63 + 35 (i^{3} {\sqrt{3}}^{3}) + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.13

Factor out $i^{2}$ .

$x = 1 + 7 i \sqrt{3} - 63 + 35 (i^{2} \cdot i {\sqrt{3}}^{3}) + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.14

Rewrite $i^{2}$ as $- 1$ .

$x = 1 + 7 i \sqrt{3} - 63 + 35 (- 1 \cdot i {\sqrt{3}}^{3}) + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.15

Rewrite $- 1 i$ as $- i$ .

$x = 1 + 7 i \sqrt{3} - 63 + 35 (- i {\sqrt{3}}^{3}) + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.16

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$x = 1 + 7 i \sqrt{3} - 63 + 35 (- i \sqrt{3^{3}}) + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.17

Raise $3$ to the power of $3$ .

$x = 1 + 7 i \sqrt{3} - 63 + 35 (- i \sqrt{27}) + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.18

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 5.2.5.2.2.1.2.1.18.1

Factor $9$ out of $27$ .

$x = 1 + 7 i \sqrt{3} - 63 + 35 (- i \sqrt{9 (3)}) + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.18.2

Rewrite $9$ as $3^{2}$ .

$x = 1 + 7 i \sqrt{3} - 63 + 35 (- i \sqrt{3^{2} \cdot 3}) + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.19

Pull terms out from under the radical.

$x = 1 + 7 i \sqrt{3} - 63 + 35 (- i (3 \sqrt{3})) + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.20

Multiply $3$ by $- 1$ .

$x = 1 + 7 i \sqrt{3} - 63 + 35 (- 3 i \sqrt{3}) + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.21

Multiply $- 3$ by $35$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 (i \sqrt{3}) + 35 \cdot 1^{3} {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.22

One to any power is one.

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 \cdot 1 {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.23

Multiply $35$ by $1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 {(i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.24

Apply the product rule to $i \sqrt{3}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 (i^{4} {\sqrt{3}}^{4}) + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.25

Rewrite $i^{4}$ as $1$ .

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Step 5.2.5.2.2.1.2.1.25.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 ({(i^{2})}^{2} {\sqrt{3}}^{4}) + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.25.2

Rewrite $i^{2}$ as $- 1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 ({(- 1)}^{2} {\sqrt{3}}^{4}) + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.25.3

Raise $- 1$ to the power of $2$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 (1 {\sqrt{3}}^{4}) + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.26

Multiply ${\sqrt{3}}^{4}$ by $1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 {\sqrt{3}}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.27

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 5.2.5.2.2.1.2.1.27.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 {(3^{\frac{1}{2}})}^{4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.27.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 \cdot 3^{\frac{1}{2} \cdot 4} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.27.3

Combine $\frac{1}{2}$ and $4$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 \cdot 3^{\frac{4}{2}} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.27.4

Cancel the common factor of $4$ and $2$ .

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Step 5.2.5.2.2.1.2.1.27.4.1

Factor $2$ out of $4$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 \cdot 3^{\frac{2 \cdot 2}{2}} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.27.4.2

Cancel the common factors.

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Step 5.2.5.2.2.1.2.1.27.4.2.1

Factor $2$ out of $2$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 \cdot 3^{\frac{2 \cdot 2}{2 (1)}} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.27.4.2.2

Cancel the common factor.

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 \cdot 3^{\frac{2 \cdot 2}{2 \cdot 1}} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.27.4.2.3

Rewrite the expression.

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 \cdot 3^{\frac{2}{1}} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.27.4.2.4

Divide $2$ by $1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 \cdot 3^{2} + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.28

Raise $3$ to the power of $2$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 35 \cdot 9 + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.29

Multiply $35$ by $9$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 21 \cdot 1^{2} {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.30

One to any power is one.

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 21 \cdot 1 {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.31

Multiply $21$ by $1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 21 {(i \sqrt{3})}^{5} + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.32

Apply the product rule to $i \sqrt{3}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 21 (i^{5} {\sqrt{3}}^{5}) + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.33

Factor out $i^{4}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 21 (i^{4} i {\sqrt{3}}^{5}) + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.34

Rewrite $i^{4}$ as $1$ .

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Step 5.2.5.2.2.1.2.1.34.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 21 ({(i^{2})}^{2} i {\sqrt{3}}^{5}) + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.34.2

Rewrite $i^{2}$ as $- 1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 21 ({(- 1)}^{2} i {\sqrt{3}}^{5}) + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.34.3

Raise $- 1$ to the power of $2$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 21 (1 i {\sqrt{3}}^{5}) + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.35

Multiply $i$ by $1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 21 (i {\sqrt{3}}^{5}) + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.36

Rewrite ${\sqrt{3}}^{5}$ as $\sqrt{3^{5}}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 21 (i \sqrt{3^{5}}) + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.37

Raise $3$ to the power of $5$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 21 (i \sqrt{243}) + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.38

Rewrite $243$ as $9^{2} \cdot 3$ .

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Step 5.2.5.2.2.1.2.1.38.1

Factor $81$ out of $243$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 21 (i \sqrt{81 (3)}) + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.38.2

Rewrite $81$ as $9^{2}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 21 (i \sqrt{9^{2} \cdot 3}) + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.39

Pull terms out from under the radical.

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 21 (i (9 \sqrt{3})) + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.40

Move $9$ to the left of $i$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 21 (9 \cdot i \sqrt{3}) + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.41

Multiply $9$ by $21$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 (i \sqrt{3}) + 7 \cdot 1 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.42

Multiply $7$ by $1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 {(i \sqrt{3})}^{6} + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.43

Apply the product rule to $i \sqrt{3}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 (i^{6} {\sqrt{3}}^{6}) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.44

Factor out $i^{4}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 (i^{4} i^{2} {\sqrt{3}}^{6}) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.45

Rewrite $i^{4}$ as $1$ .

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Step 5.2.5.2.2.1.2.1.45.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 ({(i^{2})}^{2} i^{2} {\sqrt{3}}^{6}) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.45.2

Rewrite $i^{2}$ as $- 1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 ({(- 1)}^{2} i^{2} {\sqrt{3}}^{6}) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.45.3

Raise $- 1$ to the power of $2$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 (1 i^{2} {\sqrt{3}}^{6}) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.46

Multiply $i^{2}$ by $1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 (i^{2} {\sqrt{3}}^{6}) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.47

Rewrite $i^{2}$ as $- 1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 (- 1 {\sqrt{3}}^{6}) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.48

Rewrite ${\sqrt{3}}^{6}$ as $3^{3}$ .

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Step 5.2.5.2.2.1.2.1.48.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 (- 1 {(3^{\frac{1}{2}})}^{6}) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.48.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 (- 1 \cdot 3^{\frac{1}{2} \cdot 6}) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.48.3

Combine $\frac{1}{2}$ and $6$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 (- 1 \cdot 3^{\frac{6}{2}}) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.48.4

Cancel the common factor of $6$ and $2$ .

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Step 5.2.5.2.2.1.2.1.48.4.1

Factor $2$ out of $6$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 (- 1 \cdot 3^{\frac{2 \cdot 3}{2}}) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.48.4.2

Cancel the common factors.

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Step 5.2.5.2.2.1.2.1.48.4.2.1

Factor $2$ out of $2$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 (- 1 \cdot 3^{\frac{2 \cdot 3}{2 (1)}}) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.48.4.2.2

Cancel the common factor.

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 (- 1 \cdot 3^{\frac{2 \cdot 3}{2 \cdot 1}}) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.48.4.2.3

Rewrite the expression.

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 (- 1 \cdot 3^{\frac{3}{1}}) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.48.4.2.4

Divide $3$ by $1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 (- 1 \cdot 3^{3}) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.49

Raise $3$ to the power of $3$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 (- 1 \cdot 27) + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.50

Multiply $7 (- 1 \cdot 27)$ .

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Step 5.2.5.2.2.1.2.1.50.1

Multiply $- 1$ by $27$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} + 7 \cdot - 27 + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.50.2

Multiply $7$ by $- 27$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 + {(i \sqrt{3})}^{7}$

Step 5.2.5.2.2.1.2.1.51

Apply the product rule to $i \sqrt{3}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 + i^{7} {\sqrt{3}}^{7}$

Step 5.2.5.2.2.1.2.1.52

Rewrite $i^{7}$ as $i^{4} (i^{2} \cdot i)$ .

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Step 5.2.5.2.2.1.2.1.52.1

Factor out $i^{4}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 + i^{4} i^{3} {\sqrt{3}}^{7}$

Step 5.2.5.2.2.1.2.1.52.2

Factor out $i^{2}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 + i^{4} (i^{2} \cdot i) {\sqrt{3}}^{7}$

Step 5.2.5.2.2.1.2.1.53

Rewrite $i^{4}$ as $1$ .

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Step 5.2.5.2.2.1.2.1.53.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 + {(i^{2})}^{2} (i^{2} \cdot i) {\sqrt{3}}^{7}$

Step 5.2.5.2.2.1.2.1.53.2

Rewrite $i^{2}$ as $- 1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 + {(- 1)}^{2} (i^{2} \cdot i) {\sqrt{3}}^{7}$

Step 5.2.5.2.2.1.2.1.53.3

Raise $- 1$ to the power of $2$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 + 1 (i^{2} \cdot i) {\sqrt{3}}^{7}$

Step 5.2.5.2.2.1.2.1.54

Multiply $i^{2} \cdot i$ by $1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 + i^{2} \cdot i {\sqrt{3}}^{7}$

Step 5.2.5.2.2.1.2.1.55

Rewrite $i^{2}$ as $- 1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 - 1 \cdot i {\sqrt{3}}^{7}$

Step 5.2.5.2.2.1.2.1.56

Rewrite $- 1 i$ as $- i$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 - i {\sqrt{3}}^{7}$

Step 5.2.5.2.2.1.2.1.57

Rewrite ${\sqrt{3}}^{7}$ as $\sqrt{3^{7}}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 - i \sqrt{3^{7}}$

Step 5.2.5.2.2.1.2.1.58

Raise $3$ to the power of $7$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 - i \sqrt{2187}$

Step 5.2.5.2.2.1.2.1.59

Rewrite $2187$ as $27^{2} \cdot 3$ .

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Step 5.2.5.2.2.1.2.1.59.1

Factor $729$ out of $2187$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 - i \sqrt{729 (3)}$

Step 5.2.5.2.2.1.2.1.59.2

Rewrite $729$ as $27^{2}$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 - i \sqrt{27^{2} \cdot 3}$

Step 5.2.5.2.2.1.2.1.60

Pull terms out from under the radical.

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 - i (27 \sqrt{3})$

Step 5.2.5.2.2.1.2.1.61

Multiply $27$ by $- 1$ .

$x = 1 + 7 i \sqrt{3} - 63 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 - 27 i \sqrt{3}$

Step 5.2.5.2.2.1.2.2

Simplify by adding terms.

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Step 5.2.5.2.2.1.2.2.1

Subtract $63$ from $1$ .

$x = 7 i \sqrt{3} - 62 - 105 i \sqrt{3} + 315 + 189 i \sqrt{3} - 189 - 27 i \sqrt{3}$

Step 5.2.5.2.2.1.2.2.2

Subtract $105 i \sqrt{3}$ from $7 i \sqrt{3}$ .

$x = - 98 i \sqrt{3} - 62 + 315 + 189 i \sqrt{3} - 189 - 27 i \sqrt{3}$

Step 5.2.5.2.2.1.2.2.3

Add $- 98 i \sqrt{3}$ and $189 i \sqrt{3}$ .

$x = 91 i \sqrt{3} - 62 + 315 - 189 - 27 i \sqrt{3}$

Step 5.2.5.2.2.1.2.2.4

Subtract $27 i \sqrt{3}$ from $91 i \sqrt{3}$ .

$x = 64 i \sqrt{3} - 62 + 315 - 189$

Step 5.2.5.2.2.1.2.2.5

Simplify the expression.

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Step 5.2.5.2.2.1.2.2.5.1

Add $- 62$ and $315$ .

$x = 64 i \sqrt{3} + 253 - 189$

Step 5.2.5.2.2.1.2.2.5.2

Subtract $189$ from $253$ .

$x = 64 i \sqrt{3} + 64$

Step 5.2.5.2.2.1.2.2.5.3

Reorder $64 i \sqrt{3}$ and $64$ .

$x = 64 + 64 i \sqrt{3}$

Step 5.2.6

Solve for $x^{\frac{1}{7}} = 1 - i \sqrt{3}$ for $x$ .

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Step 5.2.6.1

Raise each side of the equation to the power of $7$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{7}})}^{7} = {(1 - i \sqrt{3})}^{7}$

Step 5.2.6.2

Simplify the exponent.

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Step 5.2.6.2.1

Simplify the left side.

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Step 5.2.6.2.1.1

Simplify ${(x^{\frac{1}{7}})}^{7}$ .

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Step 5.2.6.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{7}})}^{7}$ .

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Step 5.2.6.2.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{7} \cdot 7} = {(1 - i \sqrt{3})}^{7}$

Step 5.2.6.2.1.1.1.2

Cancel the common factor of $7$ .

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Step 5.2.6.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{7} \cdot 7} = {(1 - i \sqrt{3})}^{7}$

Step 5.2.6.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = {(1 - i \sqrt{3})}^{7}$

Step 5.2.6.2.1.1.2

Simplify.

$x = {(1 - i \sqrt{3})}^{7}$

Step 5.2.6.2.2

Simplify the right side.

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Step 5.2.6.2.2.1

Simplify ${(1 - i \sqrt{3})}^{7}$ .

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Step 5.2.6.2.2.1.1

Use the Binomial Theorem.

$x = 1^{7} + 7 \cdot 1^{6} (- i \sqrt{3}) + 21 \cdot 1^{5} {(- i \sqrt{3})}^{2} + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2

Simplify terms.

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Step 5.2.6.2.2.1.2.1

Simplify each term.

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Step 5.2.6.2.2.1.2.1.1

One to any power is one.

$x = 1 + 7 \cdot 1^{6} (- i \sqrt{3}) + 21 \cdot 1^{5} {(- i \sqrt{3})}^{2} + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.2

One to any power is one.

$x = 1 + 7 \cdot 1 (- i \sqrt{3}) + 21 \cdot 1^{5} {(- i \sqrt{3})}^{2} + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.3

Multiply $7$ by $1$ .

$x = 1 + 7 (- i \sqrt{3}) + 21 \cdot 1^{5} {(- i \sqrt{3})}^{2} + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.4

Multiply $- 1$ by $7$ .

$x = 1 - 7 (i \sqrt{3}) + 21 \cdot 1^{5} {(- i \sqrt{3})}^{2} + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.5

One to any power is one.

$x = 1 - 7 i \sqrt{3} + 21 \cdot 1 {(- i \sqrt{3})}^{2} + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.6

Multiply $21$ by $1$ .

$x = 1 - 7 i \sqrt{3} + 21 {(- i \sqrt{3})}^{2} + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.6.2.2.1.2.1.7.1

Apply the product rule to $- i \sqrt{3}$ .

$x = 1 - 7 i \sqrt{3} + 21 ({(- i)}^{2} {\sqrt{3}}^{2}) + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.7.2

Apply the product rule to $- i$ .

$x = 1 - 7 i \sqrt{3} + 21 ({(- 1)}^{2} i^{2} {\sqrt{3}}^{2}) + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.8

Raise $- 1$ to the power of $2$ .

$x = 1 - 7 i \sqrt{3} + 21 (1 i^{2} {\sqrt{3}}^{2}) + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.9

Multiply $i^{2}$ by $1$ .

$x = 1 - 7 i \sqrt{3} + 21 (i^{2} {\sqrt{3}}^{2}) + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.10

Rewrite $i^{2}$ as $- 1$ .

$x = 1 - 7 i \sqrt{3} + 21 (- 1 {\sqrt{3}}^{2}) + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.11

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.2.6.2.2.1.2.1.11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = 1 - 7 i \sqrt{3} + 21 (- 1 {(3^{\frac{1}{2}})}^{2}) + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.11.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = 1 - 7 i \sqrt{3} + 21 (- 1 \cdot 3^{\frac{1}{2} \cdot 2}) + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.11.3

Combine $\frac{1}{2}$ and $2$ .

$x = 1 - 7 i \sqrt{3} + 21 (- 1 \cdot 3^{\frac{2}{2}}) + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.11.4

Cancel the common factor of $2$ .

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Step 5.2.6.2.2.1.2.1.11.4.1

Cancel the common factor.

Step 5.2.6.2.2.1.2.1.11.4.2

Rewrite the expression.

$x = 1 - 7 i \sqrt{3} + 21 (- 1 \cdot 3^{1}) + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.11.5

Evaluate the exponent.

$x = 1 - 7 i \sqrt{3} + 21 (- 1 \cdot 3) + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.12

Multiply $21 (- 1 \cdot 3)$ .

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Step 5.2.6.2.2.1.2.1.12.1

Multiply $- 1$ by $3$ .

$x = 1 - 7 i \sqrt{3} + 21 \cdot - 3 + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.12.2

Multiply $21$ by $- 3$ .

$x = 1 - 7 i \sqrt{3} - 63 + 35 \cdot 1^{4} {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.13

One to any power is one.

$x = 1 - 7 i \sqrt{3} - 63 + 35 \cdot 1 {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.14

Multiply $35$ by $1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 35 {(- i \sqrt{3})}^{3} + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.15

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.6.2.2.1.2.1.15.1

Apply the product rule to $- i \sqrt{3}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 35 ({(- i)}^{3} {\sqrt{3}}^{3}) + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.15.2

Apply the product rule to $- i$ .

$x = 1 - 7 i \sqrt{3} - 63 + 35 ({(- 1)}^{3} i^{3} {\sqrt{3}}^{3}) + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.16

Raise $- 1$ to the power of $3$ .

$x = 1 - 7 i \sqrt{3} - 63 + 35 (- i^{3} {\sqrt{3}}^{3}) + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.17

Factor out $i^{2}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 35 (- (i^{2} \cdot i) {\sqrt{3}}^{3}) + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.18

Rewrite $i^{2}$ as $- 1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 35 (- (- 1 \cdot i) {\sqrt{3}}^{3}) + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.19

Rewrite $- 1 i$ as $- i$ .

$x = 1 - 7 i \sqrt{3} - 63 + 35 (- - i {\sqrt{3}}^{3}) + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.20

Multiply $- 1$ by $- 1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 35 (1 i {\sqrt{3}}^{3}) + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.21

Multiply $i$ by $1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 35 (i {\sqrt{3}}^{3}) + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.22

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 35 (i \sqrt{3^{3}}) + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.23

Raise $3$ to the power of $3$ .

$x = 1 - 7 i \sqrt{3} - 63 + 35 (i \sqrt{27}) + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.24

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 5.2.6.2.2.1.2.1.24.1

Factor $9$ out of $27$ .

$x = 1 - 7 i \sqrt{3} - 63 + 35 (i \sqrt{9 (3)}) + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.24.2

Rewrite $9$ as $3^{2}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 35 (i \sqrt{3^{2} \cdot 3}) + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.25

Pull terms out from under the radical.

$x = 1 - 7 i \sqrt{3} - 63 + 35 (i (3 \sqrt{3})) + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.26

Move $3$ to the left of $i$ .

$x = 1 - 7 i \sqrt{3} - 63 + 35 (3 \cdot i \sqrt{3}) + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.27

Multiply $3$ by $35$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 (i \sqrt{3}) + 35 \cdot 1^{3} {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.28

One to any power is one.

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 \cdot 1 {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.29

Multiply $35$ by $1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 {(- i \sqrt{3})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.30

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.6.2.2.1.2.1.30.1

Apply the product rule to $- i \sqrt{3}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 ({(- i)}^{4} {\sqrt{3}}^{4}) + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.30.2

Apply the product rule to $- i$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 ({(- 1)}^{4} i^{4} {\sqrt{3}}^{4}) + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.31

Raise $- 1$ to the power of $4$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 (1 i^{4} {\sqrt{3}}^{4}) + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.32

Multiply $i^{4}$ by $1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 (i^{4} {\sqrt{3}}^{4}) + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.33

Rewrite $i^{4}$ as $1$ .

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Step 5.2.6.2.2.1.2.1.33.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 ({(i^{2})}^{2} {\sqrt{3}}^{4}) + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.33.2

Rewrite $i^{2}$ as $- 1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 ({(- 1)}^{2} {\sqrt{3}}^{4}) + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.33.3

Raise $- 1$ to the power of $2$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 (1 {\sqrt{3}}^{4}) + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.34

Multiply ${\sqrt{3}}^{4}$ by $1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 {\sqrt{3}}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.35

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 5.2.6.2.2.1.2.1.35.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 {(3^{\frac{1}{2}})}^{4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.35.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 \cdot 3^{\frac{1}{2} \cdot 4} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.35.3

Combine $\frac{1}{2}$ and $4$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 \cdot 3^{\frac{4}{2}} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.35.4

Cancel the common factor of $4$ and $2$ .

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Step 5.2.6.2.2.1.2.1.35.4.1

Factor $2$ out of $4$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 \cdot 3^{\frac{2 \cdot 2}{2}} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.35.4.2

Cancel the common factors.

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Step 5.2.6.2.2.1.2.1.35.4.2.1

Factor $2$ out of $2$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 \cdot 3^{\frac{2 \cdot 2}{2 (1)}} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.35.4.2.2

Cancel the common factor.

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 \cdot 3^{\frac{2 \cdot 2}{2 \cdot 1}} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.35.4.2.3

Rewrite the expression.

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 \cdot 3^{\frac{2}{1}} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.35.4.2.4

Divide $2$ by $1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 \cdot 3^{2} + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.36

Raise $3$ to the power of $2$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 35 \cdot 9 + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.37

Multiply $35$ by $9$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 \cdot 1^{2} {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.38

One to any power is one.

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 \cdot 1 {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.39

Multiply $21$ by $1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 {(- i \sqrt{3})}^{5} + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.40

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.6.2.2.1.2.1.40.1

Apply the product rule to $- i \sqrt{3}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 ({(- i)}^{5} {\sqrt{3}}^{5}) + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.40.2

Apply the product rule to $- i$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 ({(- 1)}^{5} i^{5} {\sqrt{3}}^{5}) + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.41

Raise $- 1$ to the power of $5$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 (- i^{5} {\sqrt{3}}^{5}) + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.42

Factor out $i^{4}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 (- (i^{4} i) {\sqrt{3}}^{5}) + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.43

Rewrite $i^{4}$ as $1$ .

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Step 5.2.6.2.2.1.2.1.43.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 (- ({(i^{2})}^{2} i) {\sqrt{3}}^{5}) + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.43.2

Rewrite $i^{2}$ as $- 1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 (- ({(- 1)}^{2} i) {\sqrt{3}}^{5}) + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.43.3

Raise $- 1$ to the power of $2$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 (- (1 i) {\sqrt{3}}^{5}) + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.44

Multiply $i$ by $1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 (- i {\sqrt{3}}^{5}) + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.45

Rewrite ${\sqrt{3}}^{5}$ as $\sqrt{3^{5}}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 (- i \sqrt{3^{5}}) + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.46

Raise $3$ to the power of $5$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 (- i \sqrt{243}) + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.47

Rewrite $243$ as $9^{2} \cdot 3$ .

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Step 5.2.6.2.2.1.2.1.47.1

Factor $81$ out of $243$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 (- i \sqrt{81 (3)}) + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.47.2

Rewrite $81$ as $9^{2}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 (- i \sqrt{9^{2} \cdot 3}) + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.48

Pull terms out from under the radical.

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 (- i (9 \sqrt{3})) + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.49

Multiply $9$ by $- 1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 + 21 (- 9 i \sqrt{3}) + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.50

Multiply $- 9$ by $21$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 (i \sqrt{3}) + 7 \cdot 1 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.51

Multiply $7$ by $1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 {(- i \sqrt{3})}^{6} + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.52

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.6.2.2.1.2.1.52.1

Apply the product rule to $- i \sqrt{3}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 ({(- i)}^{6} {\sqrt{3}}^{6}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.52.2

Apply the product rule to $- i$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 ({(- 1)}^{6} i^{6} {\sqrt{3}}^{6}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.53

Raise $- 1$ to the power of $6$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 (1 i^{6} {\sqrt{3}}^{6}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.54

Multiply $i^{6}$ by $1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 (i^{6} {\sqrt{3}}^{6}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.55

Factor out $i^{4}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 (i^{4} i^{2} {\sqrt{3}}^{6}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.56

Rewrite $i^{4}$ as $1$ .

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Step 5.2.6.2.2.1.2.1.56.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 ({(i^{2})}^{2} i^{2} {\sqrt{3}}^{6}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.56.2

Rewrite $i^{2}$ as $- 1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 ({(- 1)}^{2} i^{2} {\sqrt{3}}^{6}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.56.3

Raise $- 1$ to the power of $2$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 (1 i^{2} {\sqrt{3}}^{6}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.57

Multiply $i^{2}$ by $1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 (i^{2} {\sqrt{3}}^{6}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.58

Rewrite $i^{2}$ as $- 1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 (- 1 {\sqrt{3}}^{6}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.59

Rewrite ${\sqrt{3}}^{6}$ as $3^{3}$ .

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Step 5.2.6.2.2.1.2.1.59.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 (- 1 {(3^{\frac{1}{2}})}^{6}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.59.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 (- 1 \cdot 3^{\frac{1}{2} \cdot 6}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.59.3

Combine $\frac{1}{2}$ and $6$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 (- 1 \cdot 3^{\frac{6}{2}}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.59.4

Cancel the common factor of $6$ and $2$ .

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Step 5.2.6.2.2.1.2.1.59.4.1

Factor $2$ out of $6$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 (- 1 \cdot 3^{\frac{2 \cdot 3}{2}}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.59.4.2

Cancel the common factors.

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Step 5.2.6.2.2.1.2.1.59.4.2.1

Factor $2$ out of $2$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 (- 1 \cdot 3^{\frac{2 \cdot 3}{2 (1)}}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.59.4.2.2

Cancel the common factor.

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 (- 1 \cdot 3^{\frac{2 \cdot 3}{2 \cdot 1}}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.59.4.2.3

Rewrite the expression.

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 (- 1 \cdot 3^{\frac{3}{1}}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.59.4.2.4

Divide $3$ by $1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 (- 1 \cdot 3^{3}) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.60

Raise $3$ to the power of $3$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 (- 1 \cdot 27) + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.61

Multiply $7 (- 1 \cdot 27)$ .

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Step 5.2.6.2.2.1.2.1.61.1

Multiply $- 1$ by $27$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} + 7 \cdot - 27 + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.61.2

Multiply $7$ by $- 27$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 + {(- i \sqrt{3})}^{7}$

Step 5.2.6.2.2.1.2.1.62

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.6.2.2.1.2.1.62.1

Apply the product rule to $- i \sqrt{3}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 + {(- i)}^{7} {\sqrt{3}}^{7}$

Step 5.2.6.2.2.1.2.1.62.2

Apply the product rule to $- i$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 + {(- 1)}^{7} i^{7} {\sqrt{3}}^{7}$

Step 5.2.6.2.2.1.2.1.63

Raise $- 1$ to the power of $7$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 - i^{7} {\sqrt{3}}^{7}$

Step 5.2.6.2.2.1.2.1.64

Rewrite $i^{7}$ as $i^{4} (i^{2} \cdot i)$ .

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Step 5.2.6.2.2.1.2.1.64.1

Factor out $i^{4}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 - (i^{4} i^{3}) {\sqrt{3}}^{7}$

Step 5.2.6.2.2.1.2.1.64.2

Factor out $i^{2}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 - (i^{4} (i^{2} \cdot i)) {\sqrt{3}}^{7}$

Step 5.2.6.2.2.1.2.1.65

Rewrite $i^{4}$ as $1$ .

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Step 5.2.6.2.2.1.2.1.65.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 - ({(i^{2})}^{2} (i^{2} \cdot i)) {\sqrt{3}}^{7}$

Step 5.2.6.2.2.1.2.1.65.2

Rewrite $i^{2}$ as $- 1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 - ({(- 1)}^{2} (i^{2} \cdot i)) {\sqrt{3}}^{7}$

Step 5.2.6.2.2.1.2.1.65.3

Raise $- 1$ to the power of $2$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 - (1 (i^{2} \cdot i)) {\sqrt{3}}^{7}$

Step 5.2.6.2.2.1.2.1.66

Multiply $i^{2} \cdot i$ by $1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 - (i^{2} \cdot i) {\sqrt{3}}^{7}$

Step 5.2.6.2.2.1.2.1.67

Rewrite $i^{2}$ as $- 1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 - (- 1 \cdot i) {\sqrt{3}}^{7}$

Step 5.2.6.2.2.1.2.1.68

Rewrite $- 1 i$ as $- i$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 - - i {\sqrt{3}}^{7}$

Step 5.2.6.2.2.1.2.1.69

Multiply $- 1$ by $- 1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 + 1 i {\sqrt{3}}^{7}$

Step 5.2.6.2.2.1.2.1.70

Multiply $i$ by $1$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 + i {\sqrt{3}}^{7}$

Step 5.2.6.2.2.1.2.1.71

Rewrite ${\sqrt{3}}^{7}$ as $\sqrt{3^{7}}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 + i \sqrt{3^{7}}$

Step 5.2.6.2.2.1.2.1.72

Raise $3$ to the power of $7$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 + i \sqrt{2187}$

Step 5.2.6.2.2.1.2.1.73

Rewrite $2187$ as $27^{2} \cdot 3$ .

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Step 5.2.6.2.2.1.2.1.73.1

Factor $729$ out of $2187$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 + i \sqrt{729 (3)}$

Step 5.2.6.2.2.1.2.1.73.2

Rewrite $729$ as $27^{2}$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 + i \sqrt{27^{2} \cdot 3}$

Step 5.2.6.2.2.1.2.1.74

Pull terms out from under the radical.

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 + i (27 \sqrt{3})$

Step 5.2.6.2.2.1.2.1.75

Move $27$ to the left of $i$ .

$x = 1 - 7 i \sqrt{3} - 63 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 + 27 i \sqrt{3}$

Step 5.2.6.2.2.1.2.2

Simplify by adding terms.

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Step 5.2.6.2.2.1.2.2.1

Subtract $63$ from $1$ .

$x = - 7 i \sqrt{3} - 62 + 105 i \sqrt{3} + 315 - 189 i \sqrt{3} - 189 + 27 i \sqrt{3}$

Step 5.2.6.2.2.1.2.2.2

Add $- 7 i \sqrt{3}$ and $105 i \sqrt{3}$ .

$x = 98 i \sqrt{3} - 62 + 315 - 189 i \sqrt{3} - 189 + 27 i \sqrt{3}$

Step 5.2.6.2.2.1.2.2.3

Subtract $189 i \sqrt{3}$ from $98 i \sqrt{3}$ .

$x = - 91 i \sqrt{3} - 62 + 315 - 189 + 27 i \sqrt{3}$

Step 5.2.6.2.2.1.2.2.4

Add $- 91 i \sqrt{3}$ and $27 i \sqrt{3}$ .

$x = - 64 i \sqrt{3} - 62 + 315 - 189$

Step 5.2.6.2.2.1.2.2.5

Simplify the expression.

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Step 5.2.6.2.2.1.2.2.5.1

Add $- 62$ and $315$ .

$x = - 64 i \sqrt{3} + 253 - 189$

Step 5.2.6.2.2.1.2.2.5.2

Subtract $189$ from $253$ .

$x = - 64 i \sqrt{3} + 64$

Step 5.2.6.2.2.1.2.2.5.3

Reorder $- 64 i \sqrt{3}$ and $64$ .

$x = 64 - 64 i \sqrt{3}$

Step 5.2.7

List all of the solutions.

$x = 64 + 64 i \sqrt{3}, 64 - 64 i \sqrt{3}$

Step 6

The final solution is all the values that make $(x^{\frac{1}{7}} + 2) (x^{\frac{2}{7}} - 2 x^{\frac{1}{7}} + 4) = 0$ true.

$x = - 128, 64 + 64 i \sqrt{3}, 64 - 64 i \sqrt{3}$