Evaluate (27/125)^(-4/3)

${(\frac{27}{125})}^{- \frac{4}{3}}$

Step 1

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

${(\frac{125}{27})}^{\frac{4}{3}}$

Step 2

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

$\frac{125^{\frac{4}{3}}}{27^{\frac{4}{3}}}$

Step 3

Simplify the numerator.

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

$\frac{{(5^{3})}^{\frac{4}{3}}}{27^{\frac{4}{3}}}$

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

$\frac{5^{3 (\frac{4}{3})}}{27^{\frac{4}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{5^{3 (\frac{4}{3})}}{27^{\frac{4}{3}}}$

Rewrite the expression.

$\frac{5^{4}}{27^{\frac{4}{3}}}$

Raise $5$ to the power of $4$ .

$\frac{625}{27^{\frac{4}{3}}}$

Step 4

Simplify the denominator.

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

$\frac{625}{{(3^{3})}^{\frac{4}{3}}}$

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

$\frac{625}{3^{3 (\frac{4}{3})}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{625}{3^{3 (\frac{4}{3})}}$

Rewrite the expression.

$\frac{625}{3^{4}}$

Raise $3$ to the power of $4$ .

$\frac{625}{81}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{625}{81}$

Decimal Form:

$7.71604938 \dots$

Mixed Number Form:

$7 \frac{58}{81}$