Simplify sixth root of 1/(z^6*z^-8)

$\sqrt[6]{\frac{1}{z^{6} \cdot z^{- 8}}}$

Step 1

Multiply $z^{6}$ by $z^{- 8}$ by adding the exponents.

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

$\sqrt[6]{\frac{1}{z^{6 - 8}}}$

Subtract $8$ from $6$ .

$\sqrt[6]{\frac{1}{z^{- 2}}}$

Step 2

Move $z^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\sqrt[6]{z^{2}}$

Step 3

Rewrite $\sqrt[6]{z^{2}}$ as $\sqrt[3]{\sqrt{z^{2}}}$ .

$\sqrt[3]{\sqrt{z^{2}}}$

Step 4

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt[3]{z}$