Pre-Algebra Examples

${(\frac{81 a^{3} b^{9}}{a^{- b^{5}}})}^{\frac{1}{4}}$

Step 1

Cancel the common factor of $a^{3}$ and $a^{- b^{5}}$ .

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Step 1.1

Factor $a^{- b^{5}}$ out of $81 a^{3} b^{9}$ .

${(\frac{a^{- b^{5}} (81 a^{3 + b^{5}} b^{9})}{a^{- b^{5}}})}^{\frac{1}{4}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Multiply by $1$ .

${(\frac{a^{- b^{5}} (81 a^{3 + b^{5}} b^{9})}{a^{- b^{5}} \cdot 1})}^{\frac{1}{4}}$

Step 1.2.2

Cancel the common factor.

${(\frac{a^{- b^{5}} (81 a^{3 + b^{5}} b^{9})}{a^{- b^{5}} \cdot 1})}^{\frac{1}{4}}$

Step 1.2.3

Rewrite the expression.

${(\frac{81 a^{3 + b^{5}} b^{9}}{1})}^{\frac{1}{4}}$

Step 1.2.4

Divide $81 a^{3 + b^{5}} b^{9}$ by $1$ .

${(81 a^{3 + b^{5}} b^{9})}^{\frac{1}{4}}$

Step 2

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

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Step 2.1

Apply the product rule to $81 a^{3 + b^{5}} b^{9}$ .

${(81 a^{3 + b^{5}})}^{\frac{1}{4}} {(b^{9})}^{\frac{1}{4}}$

Step 2.2

Apply the product rule to $81 a^{3 + b^{5}}$ .

$81^{\frac{1}{4}} {(a^{3 + b^{5}})}^{\frac{1}{4}} {(b^{9})}^{\frac{1}{4}}$

Step 3

Simplify the expression.

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Step 3.1

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

${(3^{4})}^{\frac{1}{4}} {(a^{3 + b^{5}})}^{\frac{1}{4}} {(b^{9})}^{\frac{1}{4}}$

Step 3.2

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

$3^{4 (\frac{1}{4})} {(a^{3 + b^{5}})}^{\frac{1}{4}} {(b^{9})}^{\frac{1}{4}}$

Step 4

Cancel the common factor of $4$ .

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Step 4.1

Cancel the common factor.

$3^{4 (\frac{1}{4})} {(a^{3 + b^{5}})}^{\frac{1}{4}} {(b^{9})}^{\frac{1}{4}}$

Step 4.2

Rewrite the expression.

$3^{1} {(a^{3 + b^{5}})}^{\frac{1}{4}} {(b^{9})}^{\frac{1}{4}}$

Step 5

Evaluate the exponent.

$3 {(a^{3 + b^{5}})}^{\frac{1}{4}} {(b^{9})}^{\frac{1}{4}}$

Step 6

Multiply the exponents in ${(a^{3 + b^{5}})}^{\frac{1}{4}}$ .

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Step 6.1

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

$3 a^{(3 + b^{5}) \frac{1}{4}} {(b^{9})}^{\frac{1}{4}}$

Step 6.2

Apply the distributive property.

$3 a^{3 (\frac{1}{4}) + b^{5} \frac{1}{4}} {(b^{9})}^{\frac{1}{4}}$

Step 6.3

Combine $3$ and $\frac{1}{4}$ .

$3 a^{\frac{3}{4} + b^{5} \frac{1}{4}} {(b^{9})}^{\frac{1}{4}}$

Step 6.4

Combine $b^{5}$ and $\frac{1}{4}$ .

$3 a^{\frac{3}{4} + \frac{b^{5}}{4}} {(b^{9})}^{\frac{1}{4}}$

Step 7

Multiply the exponents in ${(b^{9})}^{\frac{1}{4}}$ .

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Step 7.1

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

$3 a^{\frac{3}{4} + \frac{b^{5}}{4}} b^{9 (\frac{1}{4})}$

Step 7.2

Combine $9$ and $\frac{1}{4}$ .

$3 a^{\frac{3}{4} + \frac{b^{5}}{4}} b^{\frac{9}{4}}$