Precalculus Examples

$x^{- \frac{2}{3}} = 36$

Step 1

Subtract $36$ from both sides of the equation.

$x^{- \frac{2}{3}} - 36 = 0$

Step 2

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

$\frac{1}{x^{\frac{2}{3}}} - 36 = 0$

Step 3

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

$\frac{1^{2}}{x^{\frac{2}{3}}} - 36 = 0$

Step 4

Rewrite $x^{\frac{2}{3}}$ as ${(x^{\frac{1}{3}})}^{2}$ .

$\frac{1^{2}}{{(x^{\frac{1}{3}})}^{2}} - 36 = 0$

Step 5

Rewrite $\frac{1^{2}}{{(x^{\frac{1}{3}})}^{2}}$ as ${(\frac{1}{x^{\frac{1}{3}}})}^{2}$ .

${(\frac{1}{x^{\frac{1}{3}}})}^{2} - 36 = 0$

Step 6

Rewrite $36$ as $6^{2}$ .

${(\frac{1}{x^{\frac{1}{3}}})}^{2} - 6^{2} = 0$

Step 7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{1}{x^{\frac{1}{3}}}$ and $b = 6$ .

$(\frac{1}{x^{\frac{1}{3}}} + 6) (\frac{1}{x^{\frac{1}{3}}} - 6) = 0$

Step 8

To write $6$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ .

$(\frac{1}{x^{\frac{1}{3}}} + \frac{6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}) (\frac{1}{x^{\frac{1}{3}}} - 6) = 0$

Step 9

Combine the numerators over the common denominator.

$\frac{1 + 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot (\frac{1}{x^{\frac{1}{3}}} - 6) = 0$

Step 10

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ .

$\frac{1 + 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot (\frac{1}{x^{\frac{1}{3}}} - 6 \cdot \frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}) = 0$

Step 11

Combine $- 6$ and $\frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ .

$\frac{1 + 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot (\frac{1}{x^{\frac{1}{3}}} + \frac{- 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}) = 0$

Step 12

Combine the numerators over the common denominator.

$\frac{1 + 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot \frac{1 - 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

Step 13

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

$\frac{1 + 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

$\frac{1 - 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

Step 14

Set $\frac{1 + 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ equal to $0$ and solve for $x$ .

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Step 14.1

Set $\frac{1 + 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ equal to $0$ .

$\frac{1 + 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

Step 14.2

Solve $\frac{1 + 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$ for $x$ .

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Step 14.2.1

Set the numerator equal to zero.

$1 + 6 x^{\frac{1}{3}} = 0$

Step 14.2.2

Solve the equation for $x$ .

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Step 14.2.2.1

Subtract $1$ from both sides of the equation.

$6 x^{\frac{1}{3}} = - 1$

Step 14.2.2.2

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

${(6 x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 14.2.2.3

Simplify the exponent.

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Step 14.2.2.3.1

Simplify the left side.

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Step 14.2.2.3.1.1

Simplify ${(6 x^{\frac{1}{3}})}^{3}$ .

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Step 14.2.2.3.1.1.1

Apply the product rule to $6 x^{\frac{1}{3}}$ .

$6^{3} {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 14.2.2.3.1.1.2

Raise $6$ to the power of $3$ .

$216 {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 14.2.2.3.1.1.3

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

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Step 14.2.2.3.1.1.3.1

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

$216 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 14.2.2.3.1.1.3.2

Cancel the common factor of $3$ .

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Step 14.2.2.3.1.1.3.2.1

Cancel the common factor.

$216 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 14.2.2.3.1.1.3.2.2

Rewrite the expression.

$216 x^{1} = {(- 1)}^{3}$

Step 14.2.2.3.1.1.4

Simplify.

$216 x = {(- 1)}^{3}$

Step 14.2.2.3.2

Simplify the right side.

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Step 14.2.2.3.2.1

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

$216 x = - 1$

Step 14.2.2.4

Divide each term in $216 x = - 1$ by $216$ and simplify.

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Step 14.2.2.4.1

Divide each term in $216 x = - 1$ by $216$ .

$\frac{216 x}{216} = \frac{- 1}{216}$

Step 14.2.2.4.2

Simplify the left side.

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Step 14.2.2.4.2.1

Cancel the common factor of $216$ .

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Step 14.2.2.4.2.1.1

Cancel the common factor.

$\frac{216 x}{216} = \frac{- 1}{216}$

Step 14.2.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{216}$

Step 14.2.2.4.3

Simplify the right side.

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Step 14.2.2.4.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{216}$

Step 15

Set $\frac{1 - 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ equal to $0$ and solve for $x$ .

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Step 15.1

Set $\frac{1 - 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ equal to $0$ .

$\frac{1 - 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

Step 15.2

Solve $\frac{1 - 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$ for $x$ .

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Step 15.2.1

Set the numerator equal to zero.

$1 - 6 x^{\frac{1}{3}} = 0$

Step 15.2.2

Solve the equation for $x$ .

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Step 15.2.2.1

Subtract $1$ from both sides of the equation.

$- 6 x^{\frac{1}{3}} = - 1$

Step 15.2.2.2

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

${(- 6 x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 15.2.2.3

Simplify the exponent.

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Step 15.2.2.3.1

Simplify the left side.

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Step 15.2.2.3.1.1

Simplify ${(- 6 x^{\frac{1}{3}})}^{3}$ .

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Step 15.2.2.3.1.1.1

Apply the product rule to $- 6 x^{\frac{1}{3}}$ .

${(- 6)}^{3} {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 15.2.2.3.1.1.2

Raise $- 6$ to the power of $3$ .

$- 216 {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 15.2.2.3.1.1.3

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

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Step 15.2.2.3.1.1.3.1

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

$- 216 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 15.2.2.3.1.1.3.2

Cancel the common factor of $3$ .

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Step 15.2.2.3.1.1.3.2.1

Cancel the common factor.

$- 216 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 15.2.2.3.1.1.3.2.2

Rewrite the expression.

$- 216 x^{1} = {(- 1)}^{3}$

Step 15.2.2.3.1.1.4

Simplify.

$- 216 x = {(- 1)}^{3}$

Step 15.2.2.3.2

Simplify the right side.

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Step 15.2.2.3.2.1

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

$- 216 x = - 1$

Step 15.2.2.4

Divide each term in $- 216 x = - 1$ by $- 216$ and simplify.

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Step 15.2.2.4.1

Divide each term in $- 216 x = - 1$ by $- 216$ .

$\frac{- 216 x}{- 216} = \frac{- 1}{- 216}$

Step 15.2.2.4.2

Simplify the left side.

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Step 15.2.2.4.2.1

Cancel the common factor of $- 216$ .

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Step 15.2.2.4.2.1.1

Cancel the common factor.

$\frac{- 216 x}{- 216} = \frac{- 1}{- 216}$

Step 15.2.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 216}$

Step 15.2.2.4.3

Simplify the right side.

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Step 15.2.2.4.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{216}$

Step 16

The final solution is all the values that make $\frac{1 + 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot \frac{1 - 6 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$ true.

$x = - \frac{1}{216}, \frac{1}{216}$