Evaluate ((3^-2)^5)/((27^-2)^2)

$\frac{{(3^{- 2})}^{5}}{{(27^{- 2})}^{2}}$

Simplify the numerator.

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

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

$\frac{3^{- 2 \cdot 5}}{{(27^{- 2})}^{2}}$

Multiply $- 2$ by $5$ .

$\frac{3^{- 10}}{{(27^{- 2})}^{2}}$

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

$\frac{\frac{1}{3^{10}}}{{(27^{- 2})}^{2}}$

Raise $3$ to the power of $10$ .

$\frac{\frac{1}{59049}}{{(27^{- 2})}^{2}}$

Simplify the denominator.

Multiply the exponents in ${(27^{- 2})}^{2}$ .

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

$\frac{\frac{1}{59049}}{27^{- 2 \cdot 2}}$

Multiply $- 2$ by $2$ .

$\frac{\frac{1}{59049}}{27^{- 4}}$

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

$\frac{\frac{1}{59049}}{\frac{1}{27^{4}}}$

Raise $27$ to the power of $4$ .

$\frac{\frac{1}{59049}}{\frac{1}{531441}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{59049} \cdot 531441$

Cancel the common factor of $59049$ .

Factor $59049$ out of $531441$ .

$\frac{1}{59049} \cdot (59049 (9))$

Cancel the common factor.

$\frac{1}{59049} \cdot (59049 \cdot 9)$

Rewrite the expression.

$9$