Evaluate ((2^5)^-3)/((2^-2)^2)

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

Move ${(2^{5})}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Simplify the denominator.

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

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

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

Multiply $- 2$ by $2$ .

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

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

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

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

$\frac{1}{\frac{1}{16} {(2^{5})}^{3}}$

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

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

$\frac{1}{\frac{1}{16} \cdot 2^{5 \cdot 3}}$

Multiply $5$ by $3$ .

$\frac{1}{\frac{1}{16} \cdot 2^{15}}$

Raise $2$ to the power of $15$ .

$\frac{1}{\frac{1}{16} \cdot 32768}$

Combine fractions.

Combine $\frac{1}{16}$ and $32768$ .

$\frac{1}{\frac{32768}{16}}$

Divide $32768$ by $16$ .

$\frac{1}{2048}$

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2048}$

Decimal Form:

$0.00048828 \dots$