Precalculus Examples

$x^{- \frac{2}{3}} = 4$

Step 1

Raise each side of the equation to the power of $- \frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(x^{- \frac{2}{3}})}^{- \frac{3}{2}} = \pm 4^{- \frac{3}{2}}$

Step 2

Simplify the exponent.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify ${(x^{- \frac{2}{3}})}^{- \frac{3}{2}}$ .

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Step 2.1.1.1

Multiply the exponents in ${(x^{- \frac{2}{3}})}^{- \frac{3}{2}}$ .

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Step 2.1.1.1.1

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

$x^{- \frac{2}{3} (- \frac{3}{2})} = \pm 4^{- \frac{3}{2}}$

Step 2.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.1.2.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$x^{\frac{- 2}{3} (- \frac{3}{2})} = \pm 4^{- \frac{3}{2}}$

Step 2.1.1.1.2.2

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$x^{\frac{- 2}{3} \cdot \frac{- 3}{2}} = \pm 4^{- \frac{3}{2}}$

Step 2.1.1.1.2.3

Factor $2$ out of $- 2$ .

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

Step 2.1.1.1.2.4

Cancel the common factor.

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

Step 2.1.1.1.2.5

Rewrite the expression.

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

Step 2.1.1.1.3

Cancel the common factor of $3$ .

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Step 2.1.1.1.3.1

Factor $3$ out of $- 3$ .

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

Step 2.1.1.1.3.2

Cancel the common factor.

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

Step 2.1.1.1.3.3

Rewrite the expression.

$x^{- - 1} = \pm 4^{- \frac{3}{2}}$

Step 2.1.1.1.4

Multiply $- 1$ by $- 1$ .

$x^{1} = \pm 4^{- \frac{3}{2}}$

Step 2.1.1.2

Simplify.

$x = \pm 4^{- \frac{3}{2}}$

Step 2.2

Simplify the right side.

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Step 2.2.1

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

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Step 2.2.1.1

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

$x = \pm \frac{1}{4^{\frac{3}{2}}}$

Step 2.2.1.2

Simplify the denominator.

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Step 2.2.1.2.1

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

$x = \pm \frac{1}{{(2^{2})}^{\frac{3}{2}}}$

Step 2.2.1.2.2

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

$x = \pm \frac{1}{2^{2 (\frac{3}{2})}}$

Step 2.2.1.2.3

Cancel the common factor of $2$ .

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Step 2.2.1.2.3.1

Cancel the common factor.

$x = \pm \frac{1}{2^{2 (\frac{3}{2})}}$

Step 2.2.1.2.3.2

Rewrite the expression.

$x = \pm \frac{1}{2^{3}}$

Step 2.2.1.2.4

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

$x = \pm \frac{1}{8}$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{1}{8}$

Step 3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{1}{8}$

Step 3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{1}{8}, - \frac{1}{8}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{8}, - \frac{1}{8}$

Decimal Form:

$x = 0.125, - 0.125$