Simplify (z/(z^-6))^(1/28)

${(\frac{z}{z^{- 6}})}^{\frac{1}{28}}$

Step 1

Move $z^{- 6}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(z \cdot z^{6})}^{\frac{1}{28}}$

Step 2

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

Multiply $z$ by $z^{6}$ .

Raise $z$ to the power of $1$ .

${(z^{1} z^{6})}^{\frac{1}{28}}$

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

${(z^{1 + 6})}^{\frac{1}{28}}$

Add $1$ and $6$ .

${(z^{7})}^{\frac{1}{28}}$

Step 3

Multiply the exponents in ${(z^{7})}^{\frac{1}{28}}$ .

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

$z^{7 (\frac{1}{28})}$

Cancel the common factor of $7$ .

Factor $7$ out of $28$ .

$z^{7 \frac{1}{7 (4)}}$

Cancel the common factor.

$z^{7 \frac{1}{7 \cdot 4}}$

Rewrite the expression.

$z^{\frac{1}{4}}$