Precalculus Examples

${(4 n)}^{\frac{3}{2}} = 512$

Step 1

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

${({(4 n)}^{\frac{3}{2}})}^{\frac{2}{3}} = 512^{\frac{2}{3}}$

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 ${({(4 n)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 2.1.1.1

Multiply the exponents in ${({(4 n)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 2.1.1.1.1

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

${(4 n)}^{\frac{3}{2} \cdot \frac{2}{3}} = 512^{\frac{2}{3}}$

Step 2.1.1.1.2

Cancel the common factor of $3$ .

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Step 2.1.1.1.2.1

Cancel the common factor.

${(4 n)}^{\frac{3}{2} \cdot \frac{2}{3}} = 512^{\frac{2}{3}}$

Step 2.1.1.1.2.2

Rewrite the expression.

${(4 n)}^{\frac{1}{2} \cdot 2} = 512^{\frac{2}{3}}$

Step 2.1.1.1.3

Cancel the common factor of $2$ .

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Step 2.1.1.1.3.1

Cancel the common factor.

${(4 n)}^{\frac{1}{2} \cdot 2} = 512^{\frac{2}{3}}$

Step 2.1.1.1.3.2

Rewrite the expression.

${(4 n)}^{1} = 512^{\frac{2}{3}}$

Step 2.1.1.2

Simplify.

$4 n = 512^{\frac{2}{3}}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $512^{\frac{2}{3}}$ .

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Step 2.2.1.1

Simplify the expression.

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Step 2.2.1.1.1

Rewrite $512$ as $8^{3}$ .

$4 n = {(8^{3})}^{\frac{2}{3}}$

Step 2.2.1.1.2

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

$4 n = 8^{3 (\frac{2}{3})}$

Step 2.2.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.2.1

Cancel the common factor.

$4 n = 8^{3 (\frac{2}{3})}$

Step 2.2.1.2.2

Rewrite the expression.

$4 n = 8^{2}$

Step 2.2.1.3

Raise $8$ to the power of $2$ .

$4 n = 64$

Step 3

Divide each term in $4 n = 64$ by $4$ and simplify.

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Step 3.1

Divide each term in $4 n = 64$ by $4$ .

$\frac{4 n}{4} = \frac{64}{4}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $4$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{4 n}{4} = \frac{64}{4}$

Step 3.2.1.2

Divide $n$ by $1$ .

$n = \frac{64}{4}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $64$ by $4$ .

$n = 16$