Evaluate cube root of 1/7

$\sqrt[3]{\frac{1}{7}}$

Step 1

Rewrite $\sqrt[3]{\frac{1}{7}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{7}}$ .

$\frac{\sqrt[3]{1}}{\sqrt[3]{7}}$

Step 2

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[3]{7}}$

Step 3

Multiply $\frac{1}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$\frac{1}{\sqrt[3]{7}} \cdot \frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$\frac{{\sqrt[3]{7}}^{2}}{\sqrt[3]{7} {\sqrt[3]{7}}^{2}}$

Raise $\sqrt[3]{7}$ to the power of $1$ .

$\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1} {\sqrt[3]{7}}^{2}}$

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

$\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{3}}$

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$\frac{{\sqrt[3]{7}}^{2}}{{(7^{\frac{1}{3}})}^{3}}$

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

$\frac{{\sqrt[3]{7}}^{2}}{7^{\frac{1}{3} \cdot 3}}$

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

$\frac{{\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{{\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{{\sqrt[3]{7}}^{2}}{7^{1}}$

Evaluate the exponent.

$\frac{{\sqrt[3]{7}}^{2}}{7}$

Step 5

Simplify the numerator.

Rewrite ${\sqrt[3]{7}}^{2}$ as $\sqrt[3]{7^{2}}$ .

$\frac{\sqrt[3]{7^{2}}}{7}$

Raise $7$ to the power of $2$ .

$\frac{\sqrt[3]{49}}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[3]{49}}{7}$

Decimal Form:

$0.52275795 \dots$

Evaluate cube root of 1/7

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