Algebra Examples

${(32 x^{- 35} y^{- 25})}^{- \frac{4}{5}}$

Step 1

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

${(32 \frac{1}{x^{35}} y^{- 25})}^{- \frac{4}{5}}$

Step 2

Combine $32$ and $\frac{1}{x^{35}}$ .

${(\frac{32}{x^{35}} y^{- 25})}^{- \frac{4}{5}}$

Step 3

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

${(\frac{32}{x^{35}} \cdot \frac{1}{y^{25}})}^{- \frac{4}{5}}$

Step 4

Combine.

${(\frac{32 \cdot 1}{x^{35} y^{25}})}^{- \frac{4}{5}}$

Step 5

Multiply $32$ by $1$ .

${(\frac{32}{x^{35} y^{25}})}^{- \frac{4}{5}}$

Step 6

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

${(\frac{x^{35} y^{25}}{32})}^{\frac{4}{5}}$

Step 7

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

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Step 7.1

Apply the product rule to $\frac{x^{35} y^{25}}{32}$ .

$\frac{{(x^{35} y^{25})}^{\frac{4}{5}}}{32^{\frac{4}{5}}}$

Step 7.2

Apply the product rule to $x^{35} y^{25}$ .

$\frac{{(x^{35})}^{\frac{4}{5}} {(y^{25})}^{\frac{4}{5}}}{32^{\frac{4}{5}}}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply the exponents in ${(x^{35})}^{\frac{4}{5}}$ .

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Step 8.1.1

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

$\frac{x^{35 (\frac{4}{5})} {(y^{25})}^{\frac{4}{5}}}{32^{\frac{4}{5}}}$

Step 8.1.2

Cancel the common factor of $5$ .

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Step 8.1.2.1

Factor $5$ out of $35$ .

$\frac{x^{5 (7) \frac{4}{5}} {(y^{25})}^{\frac{4}{5}}}{32^{\frac{4}{5}}}$

Step 8.1.2.2

Cancel the common factor.

$\frac{x^{5 \cdot 7 \frac{4}{5}} {(y^{25})}^{\frac{4}{5}}}{32^{\frac{4}{5}}}$

Step 8.1.2.3

Rewrite the expression.

$\frac{x^{7 \cdot 4} {(y^{25})}^{\frac{4}{5}}}{32^{\frac{4}{5}}}$

Step 8.1.3

Multiply $7$ by $4$ .

$\frac{x^{28} {(y^{25})}^{\frac{4}{5}}}{32^{\frac{4}{5}}}$

Step 8.2

Multiply the exponents in ${(y^{25})}^{\frac{4}{5}}$ .

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Step 8.2.1

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

$\frac{x^{28} y^{25 (\frac{4}{5})}}{32^{\frac{4}{5}}}$

Step 8.2.2

Cancel the common factor of $5$ .

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Step 8.2.2.1

Factor $5$ out of $25$ .

$\frac{x^{28} y^{5 (5) \frac{4}{5}}}{32^{\frac{4}{5}}}$

Step 8.2.2.2

Cancel the common factor.

$\frac{x^{28} y^{5 \cdot 5 \frac{4}{5}}}{32^{\frac{4}{5}}}$

Step 8.2.2.3

Rewrite the expression.

$\frac{x^{28} y^{5 \cdot 4}}{32^{\frac{4}{5}}}$

Step 8.2.3

Multiply $5$ by $4$ .

$\frac{x^{28} y^{20}}{32^{\frac{4}{5}}}$

Step 9

Simplify the denominator.

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Step 9.1

Rewrite $32$ as $2^{5}$ .

$\frac{x^{28} y^{20}}{{(2^{5})}^{\frac{4}{5}}}$

Step 9.2

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

$\frac{x^{28} y^{20}}{2^{5 (\frac{4}{5})}}$

Step 9.3

Cancel the common factor of $5$ .

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Step 9.3.1

Cancel the common factor.

$\frac{x^{28} y^{20}}{2^{5 (\frac{4}{5})}}$

Step 9.3.2

Rewrite the expression.

$\frac{x^{28} y^{20}}{2^{4}}$

Step 9.4

Raise $2$ to the power of $4$ .

$\frac{x^{28} y^{20}}{16}$