Simplify ((p^5)/(p^-3q^-4))^(1/4)

${(\frac{p^{5}}{p^{- 3} q^{- 4}})}^{\frac{1}{4}}$

Step 1

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

${(\frac{p^{5} p^{3}}{q^{- 4}})}^{\frac{1}{4}}$

Step 2

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

${(p^{5} p^{3} q^{4})}^{\frac{1}{4}}$

Step 3

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

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

${(p^{5 + 3} q^{4})}^{\frac{1}{4}}$

Add $5$ and $3$ .

${(p^{8} q^{4})}^{\frac{1}{4}}$

Step 4

Apply the product rule to $p^{8} q^{4}$ .

${(p^{8})}^{\frac{1}{4}} {(q^{4})}^{\frac{1}{4}}$

Step 5

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

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

$p^{8 (\frac{1}{4})} {(q^{4})}^{\frac{1}{4}}$

Cancel the common factor of $4$ .

Factor $4$ out of $8$ .

$p^{4 (2) \frac{1}{4}} {(q^{4})}^{\frac{1}{4}}$

Cancel the common factor.

$p^{4 \cdot 2 \frac{1}{4}} {(q^{4})}^{\frac{1}{4}}$

Rewrite the expression.

$p^{2} {(q^{4})}^{\frac{1}{4}}$

Step 6

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

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

$p^{2} q^{4 (\frac{1}{4})}$

Cancel the common factor of $4$ .

Cancel the common factor.

$p^{2} q^{4 (\frac{1}{4})}$

Rewrite the expression.

$p^{2} q^{1}$

Step 7

Simplify.

$p^{2} q$