Evaluate ( cube root of 1/5)^3

${(\sqrt[3]{\frac{1}{5}})}^{3}$

Step 1

Rewrite $\sqrt[3]{\frac{1}{5}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{5}}$ .

${(\frac{\sqrt[3]{1}}{\sqrt[3]{5}})}^{3}$

Step 2

Any root of $1$ is $1$ .

${(\frac{1}{\sqrt[3]{5}})}^{3}$

Step 3

Multiply $\frac{1}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

${(\frac{1}{\sqrt[3]{5}} \cdot \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}})}^{3}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

${(\frac{{\sqrt[3]{5}}^{2}}{\sqrt[3]{5} {\sqrt[3]{5}}^{2}})}^{3}$

Raise $\sqrt[3]{5}$ to the power of $1$ .

${(\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1} {\sqrt[3]{5}}^{2}})}^{3}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1 + 2}})}^{3}$

Add $1$ and $2$ .

${(\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{3}})}^{3}$

Rewrite ${\sqrt[3]{5}}^{3}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

${(\frac{{\sqrt[3]{5}}^{2}}{{(5^{\frac{1}{3}})}^{3}})}^{3}$

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

${(\frac{{\sqrt[3]{5}}^{2}}{5^{\frac{1}{3} \cdot 3}})}^{3}$

Combine $\frac{1}{3}$ and $3$ .

${(\frac{{\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}})}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(\frac{{\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}})}^{3}$

Rewrite the expression.

${(\frac{{\sqrt[3]{5}}^{2}}{5^{1}})}^{3}$

Evaluate the exponent.

${(\frac{{\sqrt[3]{5}}^{2}}{5})}^{3}$

Step 5

Simplify the numerator.

Rewrite ${\sqrt[3]{5}}^{2}$ as ${(5^{2})}^{\frac{1}{3}}$ .

${(\frac{\sqrt[3]{5^{2}}}{5})}^{3}$

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

${(\frac{\sqrt[3]{25}}{5})}^{3}$

Step 6

Simplify terms.

Apply the product rule to $\frac{\sqrt[3]{25}}{5}$ .

$\frac{{\sqrt[3]{25}}^{3}}{5^{3}}$

Rewrite ${\sqrt[3]{25}}^{3}$ as $25$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{25}$ as $25^{\frac{1}{3}}$ .

$\frac{{(25^{\frac{1}{3}})}^{3}}{5^{3}}$

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

$\frac{25^{\frac{1}{3} \cdot 3}}{5^{3}}$

Combine $\frac{1}{3}$ and $3$ .

$\frac{25^{\frac{3}{3}}}{5^{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{25^{\frac{3}{3}}}{5^{3}}$

Rewrite the expression.

$\frac{25^{1}}{5^{3}}$

Evaluate the exponent.

$\frac{25}{5^{3}}$

Raise $5$ to the power of $3$ .

$\frac{25}{125}$

Cancel the common factor of $25$ and $125$ .

Factor $25$ out of $25$ .

$\frac{25 (1)}{125}$

Cancel the common factors.

Factor $25$ out of $125$ .

$\frac{25 \cdot 1}{25 \cdot 5}$

Cancel the common factor.

$\frac{25 \cdot 1}{25 \cdot 5}$

Rewrite the expression.

$\frac{1}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{5}$

Decimal Form:

$0.2$

Evaluate ( cube root of 1/5)^3

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