Evaluate (27/125)^(-2/3)

${(\frac{27}{125})}^{- \frac{2}{3}}$

Step 1

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

${(\frac{125}{27})}^{\frac{2}{3}}$

Step 2

Apply the product rule to $\frac{125}{27}$ .

$\frac{125^{\frac{2}{3}}}{27^{\frac{2}{3}}}$

Step 3

Simplify the numerator.

Rewrite $125$ as $5^{3}$ .

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{5^{2}}{27^{\frac{2}{3}}}$

Raise $5$ to the power of $2$ .

$\frac{25}{27^{\frac{2}{3}}}$

Step 4

Simplify the denominator.

Rewrite $27$ as $3^{3}$ .

$\frac{25}{{(3^{3})}^{\frac{2}{3}}}$

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

$\frac{25}{3^{3 (\frac{2}{3})}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{25}{3^{3 (\frac{2}{3})}}$

Rewrite the expression.

$\frac{25}{3^{2}}$

Raise $3$ to the power of $2$ .

$\frac{25}{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{25}{9}$

Decimal Form:

$2.\overline{7}$

Mixed Number Form:

$2 \frac{7}{9}$