Question

simplify.
$left(\frac{4z^{-4}}{z^{4}}\
ight)^{3}$
write your answer using only positive exponents.

Explanation:

Step1: Simplify the inner fraction

First, combine the $z$ terms in the fraction using the exponent rule $\frac{z^a}{z^b}=z^{a-b}$.
$\frac{4z^{-4}}{z^4}=4z^{-4-4}=4z^{-8}$

Step2: Apply the power of a power rule

Raise the simplified term to the 3rd power using $(ab)^n=a^n b^n$ and $(z^a)^n=z^{a\cdot n}$.
$(4z^{-8})^3=4^3 \cdot (z^{-8})^3=64z^{-24}$

Step3: Convert to positive exponents

Use the rule $z^{-n}=\frac{1}{z^n}$ to rewrite with positive exponents.
$64z^{-24}=\frac{64}{z^{24}}$

Answer:

$\frac{64}{z^{24}}$