Simplify the Radical Expression ((z^4)/(z^-5))^(1/6)

${(\frac{z^{4}}{z^{- 5}})}^{\frac{1}{6}}$

Step 1

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

${(z^{4} z^{5})}^{\frac{1}{6}}$

Step 2

Multiply $z^{4}$ by $z^{5}$ by adding the exponents.

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

${(z^{4 + 5})}^{\frac{1}{6}}$

Add $4$ and $5$ .

${(z^{9})}^{\frac{1}{6}}$

Step 3

Multiply the exponents in ${(z^{9})}^{\frac{1}{6}}$ .

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

$z^{9 (\frac{1}{6})}$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$z^{3 (3) \frac{1}{6}}$

Factor $3$ out of $6$ .

$z^{3 \cdot 3 \frac{1}{3 \cdot 2}}$

Cancel the common factor.

$z^{3 \cdot 3 \frac{1}{3 \cdot 2}}$

Rewrite the expression.

$z^{3 (\frac{1}{2})}$

Combine $3$ and $\frac{1}{2}$ .

$z^{\frac{3}{2}}$