Simplify cube root of 1/6

$\sqrt[3]{\frac{1}{6}}$

Step 1

Rewrite $\sqrt[3]{\frac{1}{6}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{6}}$ .

$\frac{\sqrt[3]{1}}{\sqrt[3]{6}}$

Step 2

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[3]{6}}$

Step 3

Multiply $\frac{1}{\sqrt[3]{6}}$ by $\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{2}}$ .

$\frac{1}{\sqrt[3]{6}} \cdot \frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{2}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt[3]{6}}$ by $\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{2}}$ .

$\frac{{\sqrt[3]{6}}^{2}}{\sqrt[3]{6} {\sqrt[3]{6}}^{2}}$

Raise $\sqrt[3]{6}$ to the power of $1$ .

$\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{1} {\sqrt[3]{6}}^{2}}$

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

$\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{3}}$

Rewrite ${\sqrt[3]{6}}^{3}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6}$ as $6^{\frac{1}{3}}$ .

$\frac{{\sqrt[3]{6}}^{2}}{{(6^{\frac{1}{3}})}^{3}}$

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

$\frac{{\sqrt[3]{6}}^{2}}{6^{\frac{1}{3} \cdot 3}}$

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

$\frac{{\sqrt[3]{6}}^{2}}{6^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{{\sqrt[3]{6}}^{2}}{6^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{{\sqrt[3]{6}}^{2}}{6^{1}}$

Evaluate the exponent.

$\frac{{\sqrt[3]{6}}^{2}}{6}$

Step 5

Simplify the numerator.

Rewrite ${\sqrt[3]{6}}^{2}$ as $\sqrt[3]{6^{2}}$ .

$\frac{\sqrt[3]{6^{2}}}{6}$

Raise $6$ to the power of $2$ .

$\frac{\sqrt[3]{36}}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[3]{36}}{6}$

Decimal Form:

$0.55032120 \dots$

Simplify cube root of 1/6

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