Algebra Examples

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

Step 1

Move $b^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Multiply $b^{3}$ by $b^{5}$ by adding the exponents.

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Step 2.1

Move $b^{5}$ .

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

Step 2.2

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

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

Step 2.3

Add $5$ and $3$ .

${(\frac{a^{3}}{a^{- 1} b^{8}})}^{\frac{1}{4}}$

Step 3

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

${(\frac{a^{3} a}{b^{8}})}^{\frac{1}{4}}$

Step 4

Multiply $a^{3}$ by $a$ by adding the exponents.

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Step 4.1

Multiply $a^{3}$ by $a$ .

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Step 4.1.1

Raise $a$ to the power of $1$ .

${(\frac{a^{3} a^{1}}{b^{8}})}^{\frac{1}{4}}$

Step 4.1.2

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

${(\frac{a^{3 + 1}}{b^{8}})}^{\frac{1}{4}}$

Step 4.2

Add $3$ and $1$ .

${(\frac{a^{4}}{b^{8}})}^{\frac{1}{4}}$

Step 5

Apply the product rule to $\frac{a^{4}}{b^{8}}$ .

$\frac{{(a^{4})}^{\frac{1}{4}}}{{(b^{8})}^{\frac{1}{4}}}$

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

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

$\frac{a^{4 (\frac{1}{4})}}{{(b^{8})}^{\frac{1}{4}}}$

Step 6.1.2

Cancel the common factor of $4$ .

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Step 6.1.2.1

Cancel the common factor.

$\frac{a^{4 (\frac{1}{4})}}{{(b^{8})}^{\frac{1}{4}}}$

Step 6.1.2.2

Rewrite the expression.

$\frac{a^{1}}{{(b^{8})}^{\frac{1}{4}}}$

Step 6.2

Simplify.

$\frac{a}{{(b^{8})}^{\frac{1}{4}}}$

Step 7

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

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Step 7.1

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

$\frac{a}{b^{8 (\frac{1}{4})}}$

Step 7.2

Cancel the common factor of $4$ .

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Step 7.2.1

Factor $4$ out of $8$ .

$\frac{a}{b^{4 (2) \frac{1}{4}}}$

Step 7.2.2

Cancel the common factor.

$\frac{a}{b^{4 \cdot 2 \frac{1}{4}}}$

Step 7.2.3

Rewrite the expression.

$\frac{a}{b^{2}}$