Evaluate (-125/27)^(-1/3)

${(- \frac{125}{27})}^{- \frac{1}{3}}$

Step 1

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

${(- \frac{27}{125})}^{\frac{1}{3}}$

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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

${(- 1)}^{\frac{1}{3}} {(\frac{27}{125})}^{\frac{1}{3}}$

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

${(- 1)}^{\frac{1}{3}} \frac{27^{\frac{1}{3}}}{125^{\frac{1}{3}}}$

Step 3

Simplify the expression.

Rewrite $- 1$ as ${(- 1)}^{3}$ .

${({(- 1)}^{3})}^{\frac{1}{3}} \frac{27^{\frac{1}{3}}}{125^{\frac{1}{3}}}$

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

${(- 1)}^{3 (\frac{1}{3})} \frac{27^{\frac{1}{3}}}{125^{\frac{1}{3}}}$

Step 4

Cancel the common factor of $3$ .

Cancel the common factor.

${(- 1)}^{3 (\frac{1}{3})} \frac{27^{\frac{1}{3}}}{125^{\frac{1}{3}}}$

Rewrite the expression.

${(- 1)}^{1} \frac{27^{\frac{1}{3}}}{125^{\frac{1}{3}}}$

Step 5

Evaluate the exponent.

$- \frac{27^{\frac{1}{3}}}{125^{\frac{1}{3}}}$

Step 6

Simplify the numerator.

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

$- \frac{{(3^{3})}^{\frac{1}{3}}}{125^{\frac{1}{3}}}$

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

$- \frac{3^{3 (\frac{1}{3})}}{125^{\frac{1}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$- \frac{3^{3 (\frac{1}{3})}}{125^{\frac{1}{3}}}$

Rewrite the expression.

$- \frac{3^{1}}{125^{\frac{1}{3}}}$

Evaluate the exponent.

$- \frac{3}{125^{\frac{1}{3}}}$

Step 7

Simplify the denominator.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$- \frac{3}{5^{1}}$

Evaluate the exponent.

$- \frac{3}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{5}$

Decimal Form:

$- 0.6$

Evaluate (-125/27)^(-1/3)

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