Precalculus Examples

${(\frac{x^{8} y^{- 4}}{16 y^{\frac{4}{3}}})}^{- \frac{1}{4}}$

Step 1

Move $y^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{x^{8}}{16 y^{\frac{4}{3}} y^{4}})}^{- \frac{1}{4}}$

Step 2

Multiply $y^{\frac{4}{3}}$ by $y^{4}$ by adding the exponents.

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Step 2.1

Move $y^{4}$ .

${(\frac{x^{8}}{16 (y^{4} y^{\frac{4}{3}})})}^{- \frac{1}{4}}$

Step 2.2

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

${(\frac{x^{8}}{16 y^{4 + \frac{4}{3}}})}^{- \frac{1}{4}}$

Step 2.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${(\frac{x^{8}}{16 y^{4 \cdot \frac{3}{3} + \frac{4}{3}}})}^{- \frac{1}{4}}$

Step 2.4

Combine $4$ and $\frac{3}{3}$ .

${(\frac{x^{8}}{16 y^{\frac{4 \cdot 3}{3} + \frac{4}{3}}})}^{- \frac{1}{4}}$

Step 2.5

Combine the numerators over the common denominator.

${(\frac{x^{8}}{16 y^{\frac{4 \cdot 3 + 4}{3}}})}^{- \frac{1}{4}}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $4$ by $3$ .

${(\frac{x^{8}}{16 y^{\frac{12 + 4}{3}}})}^{- \frac{1}{4}}$

Step 2.6.2

Add $12$ and $4$ .

${(\frac{x^{8}}{16 y^{\frac{16}{3}}})}^{- \frac{1}{4}}$

Step 3

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{16 y^{\frac{16}{3}}}{x^{8}})}^{\frac{1}{4}}$

Step 4

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

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Step 4.1

Apply the product rule to $\frac{16 y^{\frac{16}{3}}}{x^{8}}$ .

$\frac{{(16 y^{\frac{16}{3}})}^{\frac{1}{4}}}{{(x^{8})}^{\frac{1}{4}}}$

Step 4.2

Apply the product rule to $16 y^{\frac{16}{3}}$ .

$\frac{16^{\frac{1}{4}} {(y^{\frac{16}{3}})}^{\frac{1}{4}}}{{(x^{8})}^{\frac{1}{4}}}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{{(2^{4})}^{\frac{1}{4}} {(y^{\frac{16}{3}})}^{\frac{1}{4}}}{{(x^{8})}^{\frac{1}{4}}}$

Step 5.2

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

$\frac{2^{4 (\frac{1}{4})} {(y^{\frac{16}{3}})}^{\frac{1}{4}}}{{(x^{8})}^{\frac{1}{4}}}$

Step 5.3

Cancel the common factor of $4$ .

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Step 5.3.1

Cancel the common factor.

$\frac{2^{4 (\frac{1}{4})} {(y^{\frac{16}{3}})}^{\frac{1}{4}}}{{(x^{8})}^{\frac{1}{4}}}$

Step 5.3.2

Rewrite the expression.

$\frac{2^{1} {(y^{\frac{16}{3}})}^{\frac{1}{4}}}{{(x^{8})}^{\frac{1}{4}}}$

Step 5.4

Evaluate the exponent.

$\frac{2 {(y^{\frac{16}{3}})}^{\frac{1}{4}}}{{(x^{8})}^{\frac{1}{4}}}$

Step 5.5

Multiply the exponents in ${(y^{\frac{16}{3}})}^{\frac{1}{4}}$ .

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Step 5.5.1

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

$\frac{2 y^{\frac{16}{3} \cdot \frac{1}{4}}}{{(x^{8})}^{\frac{1}{4}}}$

Step 5.5.2

Cancel the common factor of $4$ .

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Step 5.5.2.1

Factor $4$ out of $16$ .

$\frac{2 y^{\frac{4 (4)}{3} \cdot \frac{1}{4}}}{{(x^{8})}^{\frac{1}{4}}}$

Step 5.5.2.2

Cancel the common factor.

$\frac{2 y^{\frac{4 \cdot 4}{3} \cdot \frac{1}{4}}}{{(x^{8})}^{\frac{1}{4}}}$

Step 5.5.2.3

Rewrite the expression.

$\frac{2 y^{\frac{4}{3}}}{{(x^{8})}^{\frac{1}{4}}}$

Step 6

Multiply the exponents in ${(x^{8})}^{\frac{1}{4}}$ .

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Step 6.1

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

$\frac{2 y^{\frac{4}{3}}}{x^{8 (\frac{1}{4})}}$

Step 6.2

Cancel the common factor of $4$ .

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Step 6.2.1

Factor $4$ out of $8$ .

$\frac{2 y^{\frac{4}{3}}}{x^{4 (2) \frac{1}{4}}}$

Step 6.2.2

Cancel the common factor.

$\frac{2 y^{\frac{4}{3}}}{x^{4 \cdot 2 \frac{1}{4}}}$

Step 6.2.3

Rewrite the expression.

$\frac{2 y^{\frac{4}{3}}}{x^{2}}$