Algebra Examples

$\frac{\sqrt[6]{4}}{\sqrt[3]{4}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt[6]{2^{2}}}{\sqrt[3]{4}}$

Step 1.2

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

$\frac{\sqrt[3]{\sqrt{2^{2}}}}{\sqrt[3]{4}}$

Step 1.3

Pull terms out from under the radical.

$\frac{\sqrt[3]{| 2 |}}{\sqrt[3]{4}}$

Step 1.4

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{\sqrt[3]{2}}{\sqrt[3]{4}}$

Step 2

Combine $\sqrt[3]{2}$ and $\sqrt[3]{4}$ into a single radical.

$\sqrt[3]{\frac{2}{4}}$

Step 3

Cancel the common factor of $2$ and $4$ .

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Step 3.1

Factor $2$ out of $2$ .

$\sqrt[3]{\frac{2 (1)}{4}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $2$ out of $4$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Rewrite $\sqrt[3]{\frac{1}{2}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{2}}$ .

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

Step 5

Any root of $1$ is $1$ .

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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Step 7.1

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

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

Step 7.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

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

Step 7.3

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

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

Step 7.4

Add $1$ and $2$ .

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

Step 7.5

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

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Step 7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

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

Step 7.5.2

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

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

Step 7.5.3

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

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

Step 7.5.4

Cancel the common factor of $3$ .

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Step 7.5.4.1

Cancel the common factor.

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

Step 7.5.4.2

Rewrite the expression.

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

Step 7.5.5

Evaluate the exponent.

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

Step 8

Simplify the numerator.

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Step 8.1

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

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

Step 8.2

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

$\frac{\sqrt[3]{4}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[3]{4}}{2}$

Decimal Form:

$0.79370052 \dots$