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

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

Step 1

Move $a^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2

Move $b^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 3

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

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

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

Add $5$ and $3$ .

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

Step 4

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

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

Step 5

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

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

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

Cancel the common factor of $4$ .

Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

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

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

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$a^{2} b^{1}$

Step 7

Simplify.

$a^{2} b$