Algebra Examples

${(\sqrt{x - 27})}^{3} = 64$

Step 1

Solve for $\sqrt{x - 27}$ .

Tap for more steps...

Step 1.1

Subtract $64$ from both sides of the equation.

${\sqrt{x - 27}}^{3} - 64 = 0$

Step 1.2

Factor the left side of the equation.

Tap for more steps...

Step 1.2.1

Rewrite $64$ as $4^{3}$ .

${\sqrt{x - 27}}^{3} - 4^{3} = 0$

Step 1.2.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = \sqrt{x - 27}$ and $b = 4$ .

$(\sqrt{x - 27} - 4) ({\sqrt{x - 27}}^{2} + \sqrt{x - 27} \cdot 4 + 4^{2}) = 0$

Step 1.2.3

Simplify.

Tap for more steps...

Step 1.2.3.1

Move $4$ to the left of $\sqrt{x - 27}$ .

$(\sqrt{x - 27} - 4) ({\sqrt{x - 27}}^{2} + 4 \sqrt{x - 27} + 4^{2}) = 0$

Step 1.2.3.2

Raise $4$ to the power of $2$ .

$(\sqrt{x - 27} - 4) ({\sqrt{x - 27}}^{2} + 4 \sqrt{x - 27} + 16) = 0$

Step 1.3

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

$\sqrt{x - 27} - 4 = 0$

${\sqrt{x - 27}}^{2} + 4 \sqrt{x - 27} + 16 = 0$

Step 1.4

Set $\sqrt{x - 27} - 4$ equal to $0$ and solve for $\sqrt{x - 27}$ .

Tap for more steps...

Step 1.4.1

Set $\sqrt{x - 27} - 4$ equal to $0$ .

$\sqrt{x - 27} - 4 = 0$

Step 1.4.2

Add $4$ to both sides of the equation.

$\sqrt{x - 27} = 4$

Step 1.5

Set ${\sqrt{x - 27}}^{2} + 4 \sqrt{x - 27} + 16$ equal to $0$ and solve for $\sqrt{x - 27}$ .

Tap for more steps...

Step 1.5.1

Set ${\sqrt{x - 27}}^{2} + 4 \sqrt{x - 27} + 16$ equal to $0$ .

${\sqrt{x - 27}}^{2} + 4 \sqrt{x - 27} + 16 = 0$

Step 1.5.2

Solve ${\sqrt{x - 27}}^{2} + 4 \sqrt{x - 27} + 16 = 0$ for $\sqrt{x - 27}$ .

Tap for more steps...

Step 1.5.2.1

Use the quadratic formula to find the solutions.

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

Step 1.5.2.2

Substitute the values $a = 1$ , $b = 4$ , and $c = 16$ into the quadratic formula and solve for $\sqrt{x - 27}$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 16)}}{2 \cdot 1}$

Step 1.5.2.3

Simplify.

Tap for more steps...

Step 1.5.2.3.1

Simplify the numerator.

Tap for more steps...

Step 1.5.2.3.1.1

Raise $4$ to the power of $2$ .

$\sqrt{x - 27} = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 1.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

Tap for more steps...

Step 1.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$\sqrt{x - 27} = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Step 1.5.2.3.1.2.2

Multiply $- 4$ by $16$ .

$\sqrt{x - 27} = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Step 1.5.2.3.1.3

Subtract $64$ from $16$ .

$\sqrt{x - 27} = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 1.5.2.3.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$\sqrt{x - 27} = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 1.5.2.3.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$\sqrt{x - 27} = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 1.5.2.3.1.6

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

$\sqrt{x - 27} = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 1.5.2.3.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

Tap for more steps...

Step 1.5.2.3.1.7.1

Factor $16$ out of $48$ .

$\sqrt{x - 27} = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 1.5.2.3.1.7.2

Rewrite $16$ as $4^{2}$ .

$\sqrt{x - 27} = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 1.5.2.3.1.8

Pull terms out from under the radical.

$\sqrt{x - 27} = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 1.5.2.3.1.9

Move $4$ to the left of $i$ .

$\sqrt{x - 27} = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 1.5.2.3.2

Multiply $2$ by $1$ .

$\sqrt{x - 27} = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Step 1.5.2.3.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$\sqrt{x - 27} = - 2 \pm 2 i \sqrt{3}$

Step 1.5.2.4

The final answer is the combination of both solutions.

$\sqrt{x - 27} = - 2 + 2 i \sqrt{3}, - 2 - 2 i \sqrt{3}$

Step 1.6

The final solution is all the values that make $(\sqrt{x - 27} - 4) ({\sqrt{x - 27}}^{2} + 4 \sqrt{x - 27} + 16) = 0$ true.

$\sqrt{x - 27} = 4, - 2 + 2 i \sqrt{3}, - 2 - 2 i \sqrt{3}$

Step 2

Solve for $x$ in $\sqrt{x - 27} = 4$ .

Tap for more steps...

Step 2.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x - 27}}^{2} = 4^{2}$

Step 2.2

Simplify each side of the equation.

Tap for more steps...

Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 27}$ as ${(x - 27)}^{\frac{1}{2}}$ .

${({(x - 27)}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Simplify ${({(x - 27)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.2.1.1

Multiply the exponents in ${({(x - 27)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.2.1.1.1

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

${(x - 27)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 2.2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.2.1.1.2.1

Cancel the common factor.

${(x - 27)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

${(x - 27)}^{1} = 4^{2}$

Step 2.2.2.1.2

Simplify.

$x - 27 = 4^{2}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Raise $4$ to the power of $2$ .

$x - 27 = 16$

Step 2.3

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 2.3.1

Add $27$ to both sides of the equation.

$x = 16 + 27$

Step 2.3.2

Add $16$ and $27$ .

$x = 43$

Step 3

Solve for $x$ in $\sqrt{x - 27} = - 2 + 2 i \sqrt{3}$ .

Tap for more steps...

Step 3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x - 27}}^{2} = {(- 2 + 2 i \sqrt{3})}^{2}$

Step 3.2

Simplify each side of the equation.

Tap for more steps...

Step 3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 27}$ as ${(x - 27)}^{\frac{1}{2}}$ .

${({(x - 27)}^{\frac{1}{2}})}^{2} = {(- 2 + 2 i \sqrt{3})}^{2}$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Simplify ${({(x - 27)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.2.1.1

Multiply the exponents in ${({(x - 27)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.2.1.1.1

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

${(x - 27)}^{\frac{1}{2} \cdot 2} = {(- 2 + 2 i \sqrt{3})}^{2}$

Step 3.2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.1.1.2.1

Cancel the common factor.

${(x - 27)}^{\frac{1}{2} \cdot 2} = {(- 2 + 2 i \sqrt{3})}^{2}$

Step 3.2.2.1.1.2.2

Rewrite the expression.

${(x - 27)}^{1} = {(- 2 + 2 i \sqrt{3})}^{2}$

Step 3.2.2.1.2

Simplify.

$x - 27 = {(- 2 + 2 i \sqrt{3})}^{2}$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Simplify ${(- 2 + 2 i \sqrt{3})}^{2}$ .

Tap for more steps...

Step 3.2.3.1.1

Rewrite ${(- 2 + 2 i \sqrt{3})}^{2}$ as $(- 2 + 2 i \sqrt{3}) (- 2 + 2 i \sqrt{3})$ .

$x - 27 = (- 2 + 2 i \sqrt{3}) (- 2 + 2 i \sqrt{3})$

Step 3.2.3.1.2

Expand $(- 2 + 2 i \sqrt{3}) (- 2 + 2 i \sqrt{3})$ using the FOIL Method.

Tap for more steps...

Step 3.2.3.1.2.1

Apply the distributive property.

$x - 27 = - 2 (- 2 + 2 i \sqrt{3}) + 2 i \sqrt{3} (- 2 + 2 i \sqrt{3})$

Step 3.2.3.1.2.2

Apply the distributive property.

$x - 27 = - 2 \cdot - 2 - 2 (2 i \sqrt{3}) + 2 i \sqrt{3} (- 2 + 2 i \sqrt{3})$

Step 3.2.3.1.2.3

Apply the distributive property.

$x - 27 = - 2 \cdot - 2 - 2 (2 i \sqrt{3}) + 2 i \sqrt{3} \cdot - 2 + 2 i \sqrt{3} (2 i \sqrt{3})$

Step 3.2.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 3.2.3.1.3.1

Simplify each term.

Tap for more steps...

Step 3.2.3.1.3.1.1

Multiply $- 2$ by $- 2$ .

$x - 27 = 4 - 2 (2 i \sqrt{3}) + 2 i \sqrt{3} \cdot - 2 + 2 i \sqrt{3} (2 i \sqrt{3})$

Step 3.2.3.1.3.1.2

Multiply $2$ by $- 2$ .

$x - 27 = 4 - 4 (i \sqrt{3}) + 2 i \sqrt{3} \cdot - 2 + 2 i \sqrt{3} (2 i \sqrt{3})$

Step 3.2.3.1.3.1.3

Multiply $- 2$ by $2$ .

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 2 i \sqrt{3} (2 i \sqrt{3})$

Step 3.2.3.1.3.1.4

Multiply $2 i \sqrt{3} (2 i \sqrt{3})$ .

Tap for more steps...

Step 3.2.3.1.3.1.4.1

Multiply $2$ by $2$ .

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 i \sqrt{3} (i \sqrt{3})$

Step 3.2.3.1.3.1.4.2

Raise $i$ to the power of $1$ .

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 (i^{1} i) \sqrt{3} \sqrt{3}$

Step 3.2.3.1.3.1.4.3

Raise $i$ to the power of $1$ .

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 (i^{1} i^{1}) \sqrt{3} \sqrt{3}$

Step 3.2.3.1.3.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 i^{1 + 1} \sqrt{3} \sqrt{3}$

Step 3.2.3.1.3.1.4.5

Add $1$ and $1$ .

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 i^{2} \sqrt{3} \sqrt{3}$

Step 3.2.3.1.3.1.4.6

Raise $\sqrt{3}$ to the power of $1$ .

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 i^{2} ({\sqrt{3}}^{1} \sqrt{3})$

Step 3.2.3.1.3.1.4.7

Raise $\sqrt{3}$ to the power of $1$ .

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 3.2.3.1.3.1.4.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 i^{2} {\sqrt{3}}^{1 + 1}$

Step 3.2.3.1.3.1.4.9

Add $1$ and $1$ .

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 i^{2} {\sqrt{3}}^{2}$

Step 3.2.3.1.3.1.5

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

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 \cdot - 1 {\sqrt{3}}^{2}$

Step 3.2.3.1.3.1.6

Multiply $4$ by $- 1$ .

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 4 {\sqrt{3}}^{2}$

Step 3.2.3.1.3.1.7

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

Tap for more steps...

Step 3.2.3.1.3.1.7.1

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

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 4 {(3^{\frac{1}{2}})}^{2}$

Step 3.2.3.1.3.1.7.2

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

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 4 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 3.2.3.1.3.1.7.3

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

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 4 \cdot 3^{\frac{2}{2}}$

Step 3.2.3.1.3.1.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.3.1.3.1.7.4.1

Cancel the common factor.

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 4 \cdot 3^{\frac{2}{2}}$

Step 3.2.3.1.3.1.7.4.2

Rewrite the expression.

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 4 \cdot 3^{1}$

Step 3.2.3.1.3.1.7.5

Evaluate the exponent.

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 4 \cdot 3$

Step 3.2.3.1.3.1.8

Multiply $- 4$ by $3$ .

$x - 27 = 4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 12$

Step 3.2.3.1.3.2

Subtract $12$ from $4$ .

$x - 27 = - 4 i \sqrt{3} - 4 i \sqrt{3} - 8$

Step 3.2.3.1.3.3

Subtract $4 i \sqrt{3}$ from $- 4 i \sqrt{3}$ .

$x - 27 = - 8 i \sqrt{3} - 8$

Step 3.2.3.1.4

Reorder $- 8 i \sqrt{3}$ and $- 8$ .

$x - 27 = - 8 - 8 i \sqrt{3}$

Step 3.3

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 3.3.1

Add $27$ to both sides of the equation.

$x = - 8 - 8 i \sqrt{3} + 27$

Step 3.3.2

Add $- 8$ and $27$ .

$x = - 8 i \sqrt{3} + 19$

Step 3.3.3

Reorder $- 8 i \sqrt{3}$ and $19$ .

$x = 19 - 8 i \sqrt{3}$

Step 4

Solve for $x$ in $\sqrt{x - 27} = - 2 - 2 i \sqrt{3}$ .

Tap for more steps...

Step 4.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x - 27}}^{2} = {(- 2 - 2 i \sqrt{3})}^{2}$

Step 4.2

Simplify each side of the equation.

Tap for more steps...

Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 27}$ as ${(x - 27)}^{\frac{1}{2}}$ .

${({(x - 27)}^{\frac{1}{2}})}^{2} = {(- 2 - 2 i \sqrt{3})}^{2}$

Step 4.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.1

Simplify ${({(x - 27)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 4.2.2.1.1

Multiply the exponents in ${({(x - 27)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 4.2.2.1.1.1

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

${(x - 27)}^{\frac{1}{2} \cdot 2} = {(- 2 - 2 i \sqrt{3})}^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.2.1.1.2.1

Cancel the common factor.

${(x - 27)}^{\frac{1}{2} \cdot 2} = {(- 2 - 2 i \sqrt{3})}^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(x - 27)}^{1} = {(- 2 - 2 i \sqrt{3})}^{2}$

Step 4.2.2.1.2

Simplify.

$x - 27 = {(- 2 - 2 i \sqrt{3})}^{2}$

Step 4.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.3.1

Simplify ${(- 2 - 2 i \sqrt{3})}^{2}$ .

Tap for more steps...

Step 4.2.3.1.1

Rewrite ${(- 2 - 2 i \sqrt{3})}^{2}$ as $(- 2 - 2 i \sqrt{3}) (- 2 - 2 i \sqrt{3})$ .

$x - 27 = (- 2 - 2 i \sqrt{3}) (- 2 - 2 i \sqrt{3})$

Step 4.2.3.1.2

Expand $(- 2 - 2 i \sqrt{3}) (- 2 - 2 i \sqrt{3})$ using the FOIL Method.

Tap for more steps...

Step 4.2.3.1.2.1

Apply the distributive property.

$x - 27 = - 2 (- 2 - 2 i \sqrt{3}) - 2 i \sqrt{3} (- 2 - 2 i \sqrt{3})$

Step 4.2.3.1.2.2

Apply the distributive property.

$x - 27 = - 2 \cdot - 2 - 2 (- 2 i \sqrt{3}) - 2 i \sqrt{3} (- 2 - 2 i \sqrt{3})$

Step 4.2.3.1.2.3

Apply the distributive property.

$x - 27 = - 2 \cdot - 2 - 2 (- 2 i \sqrt{3}) - 2 i \sqrt{3} \cdot - 2 - 2 i \sqrt{3} (- 2 i \sqrt{3})$

Step 4.2.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 4.2.3.1.3.1

Simplify each term.

Tap for more steps...

Step 4.2.3.1.3.1.1

Multiply $- 2$ by $- 2$ .

$x - 27 = 4 - 2 (- 2 i \sqrt{3}) - 2 i \sqrt{3} \cdot - 2 - 2 i \sqrt{3} (- 2 i \sqrt{3})$

Step 4.2.3.1.3.1.2

Multiply $- 2$ by $- 2$ .

$x - 27 = 4 + 4 (i \sqrt{3}) - 2 i \sqrt{3} \cdot - 2 - 2 i \sqrt{3} (- 2 i \sqrt{3})$

Step 4.2.3.1.3.1.3

Multiply $- 2$ by $- 2$ .

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 2 i \sqrt{3} (- 2 i \sqrt{3})$

Step 4.2.3.1.3.1.4

Multiply $- 2 i \sqrt{3} (- 2 i \sqrt{3})$ .

Tap for more steps...

Step 4.2.3.1.3.1.4.1

Multiply $- 2$ by $- 2$ .

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 i \sqrt{3} (i \sqrt{3})$

Step 4.2.3.1.3.1.4.2

Raise $i$ to the power of $1$ .

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 (i^{1} i) \sqrt{3} \sqrt{3}$

Step 4.2.3.1.3.1.4.3

Raise $i$ to the power of $1$ .

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 (i^{1} i^{1}) \sqrt{3} \sqrt{3}$

Step 4.2.3.1.3.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 i^{1 + 1} \sqrt{3} \sqrt{3}$

Step 4.2.3.1.3.1.4.5

Add $1$ and $1$ .

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 i^{2} \sqrt{3} \sqrt{3}$

Step 4.2.3.1.3.1.4.6

Raise $\sqrt{3}$ to the power of $1$ .

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 i^{2} ({\sqrt{3}}^{1} \sqrt{3})$

Step 4.2.3.1.3.1.4.7

Raise $\sqrt{3}$ to the power of $1$ .

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 4.2.3.1.3.1.4.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 i^{2} {\sqrt{3}}^{1 + 1}$

Step 4.2.3.1.3.1.4.9

Add $1$ and $1$ .

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 i^{2} {\sqrt{3}}^{2}$

Step 4.2.3.1.3.1.5

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

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 \cdot - 1 {\sqrt{3}}^{2}$

Step 4.2.3.1.3.1.6

Multiply $4$ by $- 1$ .

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 4 {\sqrt{3}}^{2}$

Step 4.2.3.1.3.1.7

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

Tap for more steps...

Step 4.2.3.1.3.1.7.1

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

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 4 {(3^{\frac{1}{2}})}^{2}$

Step 4.2.3.1.3.1.7.2

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

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 4 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 4.2.3.1.3.1.7.3

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

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 4 \cdot 3^{\frac{2}{2}}$

Step 4.2.3.1.3.1.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.3.1.3.1.7.4.1

Cancel the common factor.

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 4 \cdot 3^{\frac{2}{2}}$

Step 4.2.3.1.3.1.7.4.2

Rewrite the expression.

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 4 \cdot 3^{1}$

Step 4.2.3.1.3.1.7.5

Evaluate the exponent.

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 4 \cdot 3$

Step 4.2.3.1.3.1.8

Multiply $- 4$ by $3$ .

$x - 27 = 4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 12$

Step 4.2.3.1.3.2

Subtract $12$ from $4$ .

$x - 27 = 4 i \sqrt{3} + 4 i \sqrt{3} - 8$

Step 4.2.3.1.3.3

Add $4 i \sqrt{3}$ and $4 i \sqrt{3}$ .

$x - 27 = 8 i \sqrt{3} - 8$

Step 4.2.3.1.4

Reorder $8 i \sqrt{3}$ and $- 8$ .

$x - 27 = - 8 + 8 i \sqrt{3}$

Step 4.3

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 4.3.1

Add $27$ to both sides of the equation.

$x = - 8 + 8 i \sqrt{3} + 27$

Step 4.3.2

Add $- 8$ and $27$ .

$x = 8 i \sqrt{3} + 19$

Step 4.3.3

Reorder $8 i \sqrt{3}$ and $19$ .

$x = 19 + 8 i \sqrt{3}$

Step 5

List all of the solutions.

$x = 43, 19 - 8 i \sqrt{3}, 19 + 8 i \sqrt{3}$