Simplify (-1/27)^(1/3)

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

Step 1

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

Apply the product rule to $- \frac{1}{27}$ .

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

Apply the product rule to $\frac{1}{27}$ .

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

Step 2

Simplify the expression.

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

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

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

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

Step 3

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Evaluate the exponent.

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

Step 5

One to any power is one.

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

Step 6

Simplify the denominator.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$- \frac{1}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{3}$

Decimal Form:

$- 0.\overline{3}$