Precalculus Examples

${(\frac{x^{6}}{16 y^{- 4}})}^{- \frac{1}{2}}$

Step 1

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

${(\frac{x^{6} y^{4}}{16})}^{- \frac{1}{2}}$

Step 2

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

${(\frac{16}{x^{6} y^{4}})}^{\frac{1}{2}}$

Step 3

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

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Step 3.1

Apply the product rule to $\frac{16}{x^{6} y^{4}}$ .

$\frac{16^{\frac{1}{2}}}{{(x^{6} y^{4})}^{\frac{1}{2}}}$

Step 3.2

Apply the product rule to $x^{6} y^{4}$ .

$\frac{16^{\frac{1}{2}}}{{(x^{6})}^{\frac{1}{2}} {(y^{4})}^{\frac{1}{2}}}$

Step 4

Simplify the numerator.

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Step 4.1

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

$\frac{{(4^{2})}^{\frac{1}{2}}}{{(x^{6})}^{\frac{1}{2}} {(y^{4})}^{\frac{1}{2}}}$

Step 4.2

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

$\frac{4^{2 (\frac{1}{2})}}{{(x^{6})}^{\frac{1}{2}} {(y^{4})}^{\frac{1}{2}}}$

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

Cancel the common factor.

$\frac{4^{2 (\frac{1}{2})}}{{(x^{6})}^{\frac{1}{2}} {(y^{4})}^{\frac{1}{2}}}$

Step 4.3.2

Rewrite the expression.

$\frac{4^{1}}{{(x^{6})}^{\frac{1}{2}} {(y^{4})}^{\frac{1}{2}}}$

Step 4.4

Evaluate the exponent.

$\frac{4}{{(x^{6})}^{\frac{1}{2}} {(y^{4})}^{\frac{1}{2}}}$

Step 5

Simplify the denominator.

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Step 5.1

Multiply the exponents in ${(x^{6})}^{\frac{1}{2}}$ .

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Step 5.1.1

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

$\frac{4}{x^{6 (\frac{1}{2})} {(y^{4})}^{\frac{1}{2}}}$

Step 5.1.2

Cancel the common factor of $2$ .

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Step 5.1.2.1

Factor $2$ out of $6$ .

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

Step 5.1.2.2

Cancel the common factor.

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

Step 5.1.2.3

Rewrite the expression.

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

Step 5.2

Multiply the exponents in ${(y^{4})}^{\frac{1}{2}}$ .

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Step 5.2.1

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

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

Step 5.2.2

Cancel the common factor of $2$ .

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Step 5.2.2.1

Factor $2$ out of $4$ .

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

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

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