Simplify the Radical Expression (2^-4*z^-3)^5

${(2^{- 4} \cdot z^{- 3})}^{5}$

Step 1

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

${(\frac{1}{2^{4}} \cdot z^{- 3})}^{5}$

Step 2

Raise $2$ to the power of $4$ .

${(\frac{1}{16} \cdot z^{- 3})}^{5}$

Step 3

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

${(\frac{1}{16} \cdot \frac{1}{z^{3}})}^{5}$

Step 4

Combine.

${(\frac{1 \cdot 1}{16 z^{3}})}^{5}$

Step 5

Multiply $1$ by $1$ .

${(\frac{1}{16 z^{3}})}^{5}$

Step 6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{1}{16 z^{3}}$ .

$\frac{1^{5}}{{(16 z^{3})}^{5}}$

Apply the product rule to $16 z^{3}$ .

$\frac{1^{5}}{16^{5} {(z^{3})}^{5}}$

Step 7

One to any power is one.

$\frac{1}{16^{5} {(z^{3})}^{5}}$

Step 8

Simplify the denominator.

Raise $16$ to the power of $5$ .

$\frac{1}{1048576 {(z^{3})}^{5}}$

Multiply the exponents in ${(z^{3})}^{5}$ .

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

$\frac{1}{1048576 z^{3 \cdot 5}}$

Multiply $3$ by $5$ .

$\frac{1}{1048576 z^{15}}$