Simplify the Radical Expression (p^(-3/4)q^3)^(1/3)

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

Step 1

Move the negative in front of the fraction.

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

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3

Combine $\frac{1}{p^{\frac{3}{4}}}$ and $q^{3}$ .

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

Step 4

Apply the product rule to $\frac{q^{3}}{p^{\frac{3}{4}}}$ .

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

Step 5

Simplify the numerator.

Multiply the exponents in ${(q^{3})}^{\frac{1}{3}}$ .

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 6

Multiply the exponents in ${(p^{\frac{3}{4}})}^{\frac{1}{3}}$ .

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

$\frac{q}{p^{\frac{3}{4} \cdot \frac{1}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{q}{p^{\frac{3}{4} \cdot \frac{1}{3}}}$

Rewrite the expression.

$\frac{q}{p^{\frac{1}{4}}}$