Algebra Examples

$x^{- \frac{2}{3}} - 7 x^{- \frac{1}{3}} - 15 = 0$

Step 1

Simplify each term.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{\frac{2}{3}}} - 7 x^{- \frac{1}{3}} - 15 = 0$

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{\frac{2}{3}}} - 7 \frac{1}{x^{\frac{1}{3}}} - 15 = 0$

Step 1.3

Combine $- 7$ and $\frac{1}{x^{\frac{1}{3}}}$ .

$\frac{1}{x^{\frac{2}{3}}} + \frac{- 7}{x^{\frac{1}{3}}} - 15 = 0$

Step 1.4

Move the negative in front of the fraction.

$\frac{1}{x^{\frac{2}{3}}} - \frac{7}{x^{\frac{1}{3}}} - 15 = 0$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{\frac{2}{3}}, x^{\frac{1}{3}}, 1, 1$

Step 2.2

Since $x^{\frac{2}{3}}, x^{\frac{1}{3}}, 1, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1, 1$ then find LCM for the variable part $x^{\frac{2}{3}}, x^{\frac{1}{3}}$ .

Step 2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.5

The LCM of $1, 1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 2.6

The LCM of $x^{\frac{2}{3}}, x^{\frac{1}{3}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x^{\frac{2}{3}}$

Step 3

Multiply each term in $\frac{1}{x^{\frac{2}{3}}} - \frac{7}{x^{\frac{1}{3}}} - 15 = 0$ by $x^{\frac{2}{3}}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{1}{x^{\frac{2}{3}}} - \frac{7}{x^{\frac{1}{3}}} - 15 = 0$ by $x^{\frac{2}{3}}$ .

$\frac{1}{x^{\frac{2}{3}}} x^{\frac{2}{3}} - \frac{7}{x^{\frac{1}{3}}} x^{\frac{2}{3}} - 15 x^{\frac{2}{3}} = 0 x^{\frac{2}{3}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Cancel the common factor of $x^{\frac{2}{3}}$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{1}{x^{\frac{2}{3}}} x^{\frac{2}{3}} - \frac{7}{x^{\frac{1}{3}}} x^{\frac{2}{3}} - 15 x^{\frac{2}{3}} = 0 x^{\frac{2}{3}}$

Step 3.2.1.1.2

Rewrite the expression.

$1 - \frac{7}{x^{\frac{1}{3}}} x^{\frac{2}{3}} - 15 x^{\frac{2}{3}} = 0 x^{\frac{2}{3}}$

Step 3.2.1.2

Cancel the common factor of $x^{\frac{1}{3}}$ .

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Step 3.2.1.2.1

Move the leading negative in $- \frac{7}{x^{\frac{1}{3}}}$ into the numerator.

$1 + \frac{- 7}{x^{\frac{1}{3}}} x^{\frac{2}{3}} - 15 x^{\frac{2}{3}} = 0 x^{\frac{2}{3}}$

Step 3.2.1.2.2

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{2}{3}}$ .

$1 + \frac{- 7}{x^{\frac{1}{3}}} (x^{\frac{1}{3}} x^{\frac{1}{3}}) - 15 x^{\frac{2}{3}} = 0 x^{\frac{2}{3}}$

Step 3.2.1.2.3

Cancel the common factor.

$1 + \frac{- 7}{x^{\frac{1}{3}}} (x^{\frac{1}{3}} x^{\frac{1}{3}}) - 15 x^{\frac{2}{3}} = 0 x^{\frac{2}{3}}$

Step 3.2.1.2.4

Rewrite the expression.

$1 - 7 x^{\frac{1}{3}} - 15 x^{\frac{2}{3}} = 0 x^{\frac{2}{3}}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $0$ by $x^{\frac{2}{3}}$ .

$1 - 7 x^{\frac{1}{3}} - 15 x^{\frac{2}{3}} = 0$

Step 4

Solve the equation.

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Step 4.1

Find a common factor $x^{\frac{1}{3}}$ that is present in each term.

$1 - 7 x^{\frac{1}{3}} - 15 {(x^{\frac{1}{3}})}^{2}$

Step 4.2

Substitute $u$ for $x^{\frac{1}{3}}$ .

$1 - 7 u - 15 {(u)}^{2} = 0$

Step 4.3

Solve for $u$ .

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Step 4.3.1

Remove parentheses.

$1 - 7 u - 15 u^{2} = 0$

Step 4.3.2

Use the quadratic formula to find the solutions.

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

Step 4.3.3

Substitute the values $a = - 15$ , $b = - 7$ , and $c = 1$ into the quadratic formula and solve for $u$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (- 15 \cdot 1)}}{2 \cdot - 15}$

Step 4.3.4

Simplify.

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Step 4.3.4.1

Simplify the numerator.

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Step 4.3.4.1.1

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

$u = \frac{7 \pm \sqrt{49 - 4 \cdot - 15 \cdot 1}}{2 \cdot - 15}$

Step 4.3.4.1.2

Multiply $- 4 \cdot - 15 \cdot 1$ .

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Step 4.3.4.1.2.1

Multiply $- 4$ by $- 15$ .

$u = \frac{7 \pm \sqrt{49 + 60 \cdot 1}}{2 \cdot - 15}$

Step 4.3.4.1.2.2

Multiply $60$ by $1$ .

$u = \frac{7 \pm \sqrt{49 + 60}}{2 \cdot - 15}$

Step 4.3.4.1.3

Add $49$ and $60$ .

$u = \frac{7 \pm \sqrt{109}}{2 \cdot - 15}$

Step 4.3.4.2

Multiply $2$ by $- 15$ .

$u = \frac{7 \pm \sqrt{109}}{- 30}$

Step 4.3.4.3

Move the negative in front of the fraction.

$u = - \frac{7 \pm \sqrt{109}}{30}$

Step 4.3.5

The final answer is the combination of both solutions.

$u = - \frac{7 + \sqrt{109}}{30}, - \frac{7 - \sqrt{109}}{30}$

Step 4.4

Substitute $x$ for $u$ .

$x^{\frac{1}{3}} = - \frac{7 + \sqrt{109}}{30}, - \frac{7 - \sqrt{109}}{30}$

Step 4.5

Solve for $x^{\frac{1}{3}} = - \frac{7 + \sqrt{109}}{30}$ for $x$ .

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Step 4.5.1

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{3}})}^{3} = {(- \frac{7 + \sqrt{109}}{30})}^{3}$

Step 4.5.2

Simplify the exponent.

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Step 4.5.2.1

Simplify the left side.

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Step 4.5.2.1.1

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

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Step 4.5.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 4.5.2.1.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = {(- \frac{7 + \sqrt{109}}{30})}^{3}$

Step 4.5.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 4.5.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = {(- \frac{7 + \sqrt{109}}{30})}^{3}$

Step 4.5.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = {(- \frac{7 + \sqrt{109}}{30})}^{3}$

Step 4.5.2.1.1.2

Simplify.

$x = {(- \frac{7 + \sqrt{109}}{30})}^{3}$

Step 4.5.2.2

Simplify the right side.

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Step 4.5.2.2.1

Simplify ${(- \frac{7 + \sqrt{109}}{30})}^{3}$ .

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Step 4.5.2.2.1.1

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

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Step 4.5.2.2.1.1.1

Apply the product rule to $- \frac{7 + \sqrt{109}}{30}$ .

$x = {(- 1)}^{3} {(\frac{7 + \sqrt{109}}{30})}^{3}$

Step 4.5.2.2.1.1.2

Apply the product rule to $\frac{7 + \sqrt{109}}{30}$ .

$x = {(- 1)}^{3} \frac{{(7 + \sqrt{109})}^{3}}{30^{3}}$

Step 4.5.2.2.1.2

Evaluate the exponents.

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Step 4.5.2.2.1.2.1

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

$x = - \frac{{(7 + \sqrt{109})}^{3}}{30^{3}}$

Step 4.5.2.2.1.2.2

Raise $30$ to the power of $3$ .

$x = - \frac{{(7 + \sqrt{109})}^{3}}{27000}$

Step 4.5.2.2.1.3

Use the Binomial Theorem.

$x = - \frac{7^{3} + 3 \cdot 7^{2} \sqrt{109} + 3 \cdot 7 {\sqrt{109}}^{2} + {\sqrt{109}}^{3}}{27000}$

Step 4.5.2.2.1.4

Simplify terms.

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Step 4.5.2.2.1.4.1

Simplify each term.

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Step 4.5.2.2.1.4.1.1

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

$x = - \frac{343 + 3 \cdot 7^{2} \sqrt{109} + 3 \cdot 7 {\sqrt{109}}^{2} + {\sqrt{109}}^{3}}{27000}$

Step 4.5.2.2.1.4.1.2

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

$x = - \frac{343 + 3 \cdot 49 \sqrt{109} + 3 \cdot 7 {\sqrt{109}}^{2} + {\sqrt{109}}^{3}}{27000}$

Step 4.5.2.2.1.4.1.3

Multiply $3$ by $49$ .

$x = - \frac{343 + 147 \sqrt{109} + 3 \cdot 7 {\sqrt{109}}^{2} + {\sqrt{109}}^{3}}{27000}$

Step 4.5.2.2.1.4.1.4

Multiply $3$ by $7$ .

$x = - \frac{343 + 147 \sqrt{109} + 21 {\sqrt{109}}^{2} + {\sqrt{109}}^{3}}{27000}$

Step 4.5.2.2.1.4.1.5

Rewrite ${\sqrt{109}}^{2}$ as $109$ .

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Step 4.5.2.2.1.4.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{109}$ as $109^{\frac{1}{2}}$ .

$x = - \frac{343 + 147 \sqrt{109} + 21 {(109^{\frac{1}{2}})}^{2} + {\sqrt{109}}^{3}}{27000}$

Step 4.5.2.2.1.4.1.5.2

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

$x = - \frac{343 + 147 \sqrt{109} + 21 \cdot 109^{\frac{1}{2} \cdot 2} + {\sqrt{109}}^{3}}{27000}$

Step 4.5.2.2.1.4.1.5.3

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

$x = - \frac{343 + 147 \sqrt{109} + 21 \cdot 109^{\frac{2}{2}} + {\sqrt{109}}^{3}}{27000}$

Step 4.5.2.2.1.4.1.5.4

Cancel the common factor of $2$ .

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Step 4.5.2.2.1.4.1.5.4.1

Cancel the common factor.

$x = - \frac{343 + 147 \sqrt{109} + 21 \cdot 109^{\frac{2}{2}} + {\sqrt{109}}^{3}}{27000}$

Step 4.5.2.2.1.4.1.5.4.2

Rewrite the expression.

$x = - \frac{343 + 147 \sqrt{109} + 21 \cdot 109^{1} + {\sqrt{109}}^{3}}{27000}$

Step 4.5.2.2.1.4.1.5.5

Evaluate the exponent.

$x = - \frac{343 + 147 \sqrt{109} + 21 \cdot 109 + {\sqrt{109}}^{3}}{27000}$

Step 4.5.2.2.1.4.1.6

Multiply $21$ by $109$ .

$x = - \frac{343 + 147 \sqrt{109} + 2289 + {\sqrt{109}}^{3}}{27000}$

Step 4.5.2.2.1.4.1.7

Rewrite ${\sqrt{109}}^{3}$ as $\sqrt{109^{3}}$ .

$x = - \frac{343 + 147 \sqrt{109} + 2289 + \sqrt{109^{3}}}{27000}$

Step 4.5.2.2.1.4.1.8

Raise $109$ to the power of $3$ .

$x = - \frac{343 + 147 \sqrt{109} + 2289 + \sqrt{1295029}}{27000}$

Step 4.5.2.2.1.4.1.9

Rewrite $1295029$ as $109^{2} \cdot 109$ .

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Step 4.5.2.2.1.4.1.9.1

Factor $11881$ out of $1295029$ .

$x = - \frac{343 + 147 \sqrt{109} + 2289 + \sqrt{11881 (109)}}{27000}$

Step 4.5.2.2.1.4.1.9.2

Rewrite $11881$ as $109^{2}$ .

$x = - \frac{343 + 147 \sqrt{109} + 2289 + \sqrt{109^{2} \cdot 109}}{27000}$

Step 4.5.2.2.1.4.1.10

Pull terms out from under the radical.

$x = - \frac{343 + 147 \sqrt{109} + 2289 + 109 \sqrt{109}}{27000}$

Step 4.5.2.2.1.4.2

Simplify terms.

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Step 4.5.2.2.1.4.2.1

Add $343$ and $2289$ .

$x = - \frac{2632 + 147 \sqrt{109} + 109 \sqrt{109}}{27000}$

Step 4.5.2.2.1.4.2.2

Add $147 \sqrt{109}$ and $109 \sqrt{109}$ .

$x = - \frac{2632 + 256 \sqrt{109}}{27000}$

Step 4.5.2.2.1.4.2.3

Cancel the common factor of $2632 + 256 \sqrt{109}$ and $27000$ .

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Step 4.5.2.2.1.4.2.3.1

Factor $8$ out of $2632$ .

$x = - \frac{8 (329) + 256 \sqrt{109}}{27000}$

Step 4.5.2.2.1.4.2.3.2

Factor $8$ out of $256 \sqrt{109}$ .

$x = - \frac{8 (329) + 8 (32 \sqrt{109})}{27000}$

Step 4.5.2.2.1.4.2.3.3

Factor $8$ out of $8 (329) + 8 (32 \sqrt{109})$ .

$x = - \frac{8 (329 + 32 \sqrt{109})}{27000}$

Step 4.5.2.2.1.4.2.3.4

Cancel the common factors.

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Step 4.5.2.2.1.4.2.3.4.1

Factor $8$ out of $27000$ .

$x = - \frac{8 (329 + 32 \sqrt{109})}{8 \cdot 3375}$

Step 4.5.2.2.1.4.2.3.4.2

Cancel the common factor.

$x = - \frac{8 (329 + 32 \sqrt{109})}{8 \cdot 3375}$

Step 4.5.2.2.1.4.2.3.4.3

Rewrite the expression.

$x = - \frac{329 + 32 \sqrt{109}}{3375}$

Step 4.6

Solve for $x^{\frac{1}{3}} = - \frac{7 - \sqrt{109}}{30}$ for $x$ .

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Step 4.6.1

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{3}})}^{3} = {(- \frac{7 - \sqrt{109}}{30})}^{3}$

Step 4.6.2

Simplify the exponent.

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Step 4.6.2.1

Simplify the left side.

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Step 4.6.2.1.1

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

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Step 4.6.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 4.6.2.1.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = {(- \frac{7 - \sqrt{109}}{30})}^{3}$

Step 4.6.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 4.6.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = {(- \frac{7 - \sqrt{109}}{30})}^{3}$

Step 4.6.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = {(- \frac{7 - \sqrt{109}}{30})}^{3}$

Step 4.6.2.1.1.2

Simplify.

$x = {(- \frac{7 - \sqrt{109}}{30})}^{3}$

Step 4.6.2.2

Simplify the right side.

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Step 4.6.2.2.1

Simplify ${(- \frac{7 - \sqrt{109}}{30})}^{3}$ .

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Step 4.6.2.2.1.1

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

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Step 4.6.2.2.1.1.1

Apply the product rule to $- \frac{7 - \sqrt{109}}{30}$ .

$x = {(- 1)}^{3} {(\frac{7 - \sqrt{109}}{30})}^{3}$

Step 4.6.2.2.1.1.2

Apply the product rule to $\frac{7 - \sqrt{109}}{30}$ .

$x = {(- 1)}^{3} \frac{{(7 - \sqrt{109})}^{3}}{30^{3}}$

Step 4.6.2.2.1.2

Evaluate the exponents.

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Step 4.6.2.2.1.2.1

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

$x = - \frac{{(7 - \sqrt{109})}^{3}}{30^{3}}$

Step 4.6.2.2.1.2.2

Raise $30$ to the power of $3$ .

$x = - \frac{{(7 - \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.3

Use the Binomial Theorem.

$x = - \frac{7^{3} + 3 \cdot 7^{2} (- \sqrt{109}) + 3 \cdot 7 {(- \sqrt{109})}^{2} + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4

Simplify terms.

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Step 4.6.2.2.1.4.1

Simplify each term.

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Step 4.6.2.2.1.4.1.1

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

$x = - \frac{343 + 3 \cdot 7^{2} (- \sqrt{109}) + 3 \cdot 7 {(- \sqrt{109})}^{2} + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4.1.2

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

$x = - \frac{343 + 3 \cdot 49 (- \sqrt{109}) + 3 \cdot 7 {(- \sqrt{109})}^{2} + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4.1.3

Multiply $3$ by $49$ .

$x = - \frac{343 + 147 (- \sqrt{109}) + 3 \cdot 7 {(- \sqrt{109})}^{2} + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4.1.4

Multiply $- 1$ by $147$ .

$x = - \frac{343 - 147 \sqrt{109} + 3 \cdot 7 {(- \sqrt{109})}^{2} + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4.1.5

Multiply $3$ by $7$ .

$x = - \frac{343 - 147 \sqrt{109} + 21 {(- \sqrt{109})}^{2} + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4.1.6

Apply the product rule to $- \sqrt{109}$ .

$x = - \frac{343 - 147 \sqrt{109} + 21 ({(- 1)}^{2} {\sqrt{109}}^{2}) + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4.1.7

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

$x = - \frac{343 - 147 \sqrt{109} + 21 (1 {\sqrt{109}}^{2}) + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4.1.8

Multiply ${\sqrt{109}}^{2}$ by $1$ .

$x = - \frac{343 - 147 \sqrt{109} + 21 {\sqrt{109}}^{2} + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4.1.9

Rewrite ${\sqrt{109}}^{2}$ as $109$ .

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Step 4.6.2.2.1.4.1.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{109}$ as $109^{\frac{1}{2}}$ .

$x = - \frac{343 - 147 \sqrt{109} + 21 {(109^{\frac{1}{2}})}^{2} + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4.1.9.2

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

$x = - \frac{343 - 147 \sqrt{109} + 21 \cdot 109^{\frac{1}{2} \cdot 2} + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4.1.9.3

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

$x = - \frac{343 - 147 \sqrt{109} + 21 \cdot 109^{\frac{2}{2}} + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4.1.9.4

Cancel the common factor of $2$ .

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Step 4.6.2.2.1.4.1.9.4.1

Cancel the common factor.

$x = - \frac{343 - 147 \sqrt{109} + 21 \cdot 109^{\frac{2}{2}} + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4.1.9.4.2

Rewrite the expression.

$x = - \frac{343 - 147 \sqrt{109} + 21 \cdot 109^{1} + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4.1.9.5

Evaluate the exponent.

$x = - \frac{343 - 147 \sqrt{109} + 21 \cdot 109 + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4.1.10

Multiply $21$ by $109$ .

$x = - \frac{343 - 147 \sqrt{109} + 2289 + {(- \sqrt{109})}^{3}}{27000}$

Step 4.6.2.2.1.4.1.11

Apply the product rule to $- \sqrt{109}$ .

$x = - \frac{343 - 147 \sqrt{109} + 2289 + {(- 1)}^{3} {\sqrt{109}}^{3}}{27000}$

Step 4.6.2.2.1.4.1.12

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

$x = - \frac{343 - 147 \sqrt{109} + 2289 - {\sqrt{109}}^{3}}{27000}$

Step 4.6.2.2.1.4.1.13

Rewrite ${\sqrt{109}}^{3}$ as $\sqrt{109^{3}}$ .

$x = - \frac{343 - 147 \sqrt{109} + 2289 - \sqrt{109^{3}}}{27000}$

Step 4.6.2.2.1.4.1.14

Raise $109$ to the power of $3$ .

$x = - \frac{343 - 147 \sqrt{109} + 2289 - \sqrt{1295029}}{27000}$

Step 4.6.2.2.1.4.1.15

Rewrite $1295029$ as $109^{2} \cdot 109$ .

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Step 4.6.2.2.1.4.1.15.1

Factor $11881$ out of $1295029$ .

$x = - \frac{343 - 147 \sqrt{109} + 2289 - \sqrt{11881 (109)}}{27000}$

Step 4.6.2.2.1.4.1.15.2

Rewrite $11881$ as $109^{2}$ .

$x = - \frac{343 - 147 \sqrt{109} + 2289 - \sqrt{109^{2} \cdot 109}}{27000}$

Step 4.6.2.2.1.4.1.16

Pull terms out from under the radical.

$x = - \frac{343 - 147 \sqrt{109} + 2289 - (109 \sqrt{109})}{27000}$

Step 4.6.2.2.1.4.1.17

Multiply $109$ by $- 1$ .

$x = - \frac{343 - 147 \sqrt{109} + 2289 - 109 \sqrt{109}}{27000}$

Step 4.6.2.2.1.4.2

Simplify terms.

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Step 4.6.2.2.1.4.2.1

Add $343$ and $2289$ .

$x = - \frac{2632 - 147 \sqrt{109} - 109 \sqrt{109}}{27000}$

Step 4.6.2.2.1.4.2.2

Subtract $109 \sqrt{109}$ from $- 147 \sqrt{109}$ .

$x = - \frac{2632 - 256 \sqrt{109}}{27000}$

Step 4.6.2.2.1.4.2.3

Cancel the common factor of $2632 - 256 \sqrt{109}$ and $27000$ .

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Step 4.6.2.2.1.4.2.3.1

Factor $8$ out of $2632$ .

$x = - \frac{8 (329) - 256 \sqrt{109}}{27000}$

Step 4.6.2.2.1.4.2.3.2

Factor $8$ out of $- 256 \sqrt{109}$ .

$x = - \frac{8 (329) + 8 (- 32 \sqrt{109})}{27000}$

Step 4.6.2.2.1.4.2.3.3

Factor $8$ out of $8 (329) + 8 (- 32 \sqrt{109})$ .

$x = - \frac{8 (329 - 32 \sqrt{109})}{27000}$

Step 4.6.2.2.1.4.2.3.4

Cancel the common factors.

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Step 4.6.2.2.1.4.2.3.4.1

Factor $8$ out of $27000$ .

$x = - \frac{8 (329 - 32 \sqrt{109})}{8 \cdot 3375}$

Step 4.6.2.2.1.4.2.3.4.2

Cancel the common factor.

$x = - \frac{8 (329 - 32 \sqrt{109})}{8 \cdot 3375}$

Step 4.6.2.2.1.4.2.3.4.3

Rewrite the expression.

$x = - \frac{329 - 32 \sqrt{109}}{3375}$

Step 4.7

List all of the solutions.

$x = - \frac{329 + 32 \sqrt{109}}{3375}, - \frac{329 - 32 \sqrt{109}}{3375}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{329 + 32 \sqrt{109}}{3375}, - \frac{329 - 32 \sqrt{109}}{3375}$

Decimal Form:

$x = - 0.19647105 \dots, 0.00150809 \dots$