Simplify ((a^3b^-5)/(a^-1b^3))^(1/4)

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

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.

Move $b^{5}$ .

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

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

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

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.

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

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

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

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

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

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.

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

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

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 7

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

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

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

Cancel the common factor of $4$ .

Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify ((a^3b^-5)/(a^-1b^3))^(1/4)

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