Simplify the Radical Expression ((8^-5)/(2^-2))^-4

${(\frac{8^{- 5}}{2^{- 2}})}^{- 4}$

Step 1

Move $8^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{1}{2^{- 2} \cdot 8^{5}})}^{- 4}$

Step 2

Simplify the denominator.

Rewrite $8$ as $2^{3}$ .

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

Multiply the exponents in ${(2^{3})}^{5}$ .

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

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

Multiply $3$ by $5$ .

${(\frac{1}{2^{- 2} \cdot 2^{15}})}^{- 4}$

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

${(\frac{1}{2^{- 2 + 15}})}^{- 4}$

Add $- 2$ and $15$ .

${(\frac{1}{2^{13}})}^{- 4}$

Step 3

Raise $2$ to the power of $13$ .

${(\frac{1}{8192})}^{- 4}$

Step 4

Change the sign of the exponent by rewriting the base as its reciprocal.

$8192^{4}$

Step 5

Raise $8192$ to the power of $4$ .

$4503599627370500$