Algebra Examples

${(81 x^{- 12} y^{20})}^{- \frac{3}{4}}$

Step 1

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

${(81 \frac{1}{x^{12}} y^{20})}^{- \frac{3}{4}}$

Step 2

Combine $81$ and $\frac{1}{x^{12}}$ .

${(\frac{81}{x^{12}} y^{20})}^{- \frac{3}{4}}$

Step 3

Combine $\frac{81}{x^{12}}$ and $y^{20}$ .

${(\frac{81 y^{20}}{x^{12}})}^{- \frac{3}{4}}$

Step 4

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

${(\frac{x^{12}}{81 y^{20}})}^{\frac{3}{4}}$

Step 5

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

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Step 5.1

Apply the product rule to $\frac{x^{12}}{81 y^{20}}$ .

$\frac{{(x^{12})}^{\frac{3}{4}}}{{(81 y^{20})}^{\frac{3}{4}}}$

Step 5.2

Apply the product rule to $81 y^{20}$ .

$\frac{{(x^{12})}^{\frac{3}{4}}}{81^{\frac{3}{4}} {(y^{20})}^{\frac{3}{4}}}$

Step 6

Multiply the exponents in ${(x^{12})}^{\frac{3}{4}}$ .

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Step 6.1

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

$\frac{x^{12 (\frac{3}{4})}}{81^{\frac{3}{4}} {(y^{20})}^{\frac{3}{4}}}$

Step 6.2

Cancel the common factor of $4$ .

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Step 6.2.1

Factor $4$ out of $12$ .

$\frac{x^{4 (3) \frac{3}{4}}}{81^{\frac{3}{4}} {(y^{20})}^{\frac{3}{4}}}$

Step 6.2.2

Cancel the common factor.

$\frac{x^{4 \cdot 3 \frac{3}{4}}}{81^{\frac{3}{4}} {(y^{20})}^{\frac{3}{4}}}$

Step 6.2.3

Rewrite the expression.

$\frac{x^{3 \cdot 3}}{81^{\frac{3}{4}} {(y^{20})}^{\frac{3}{4}}}$

Step 6.3

Multiply $3$ by $3$ .

$\frac{x^{9}}{81^{\frac{3}{4}} {(y^{20})}^{\frac{3}{4}}}$

Step 7

Simplify the denominator.

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Step 7.1

Rewrite $81$ as $3^{4}$ .

$\frac{x^{9}}{{(3^{4})}^{\frac{3}{4}} {(y^{20})}^{\frac{3}{4}}}$

Step 7.2

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

$\frac{x^{9}}{3^{4 (\frac{3}{4})} {(y^{20})}^{\frac{3}{4}}}$

Step 7.3

Cancel the common factor of $4$ .

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Step 7.3.1

Cancel the common factor.

$\frac{x^{9}}{3^{4 (\frac{3}{4})} {(y^{20})}^{\frac{3}{4}}}$

Step 7.3.2

Rewrite the expression.

$\frac{x^{9}}{3^{3} {(y^{20})}^{\frac{3}{4}}}$

Step 7.4

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

$\frac{x^{9}}{27 {(y^{20})}^{\frac{3}{4}}}$

Step 7.5

Multiply the exponents in ${(y^{20})}^{\frac{3}{4}}$ .

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Step 7.5.1

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

$\frac{x^{9}}{27 y^{20 (\frac{3}{4})}}$

Step 7.5.2

Cancel the common factor of $4$ .

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Step 7.5.2.1

Factor $4$ out of $20$ .

$\frac{x^{9}}{27 y^{4 (5) \frac{3}{4}}}$

Step 7.5.2.2

Cancel the common factor.

$\frac{x^{9}}{27 y^{4 \cdot 5 \frac{3}{4}}}$

Step 7.5.2.3

Rewrite the expression.

$\frac{x^{9}}{27 y^{5 \cdot 3}}$

Step 7.5.3

Multiply $5$ by $3$ .

$\frac{x^{9}}{27 y^{15}}$