Algebra Examples

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

Step 1

Rewrite $\sqrt[3]{\frac{3}{4 k}}$ as $\frac{\sqrt[3]{3}}{\sqrt[3]{4 k}}$ .

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

Step 2

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{4 k}}$ by $\frac{{\sqrt[3]{4 k}}^{2}}{{\sqrt[3]{4 k}}^{2}}$ .

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

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{4 k}}$ by $\frac{{\sqrt[3]{4 k}}^{2}}{{\sqrt[3]{4 k}}^{2}}$ .

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

Step 3.2

Raise $\sqrt[3]{4 k}$ to the power of $1$ .

$\frac{\sqrt[3]{3} {\sqrt[3]{4 k}}^{2}}{{\sqrt[3]{4 k}}^{1} {\sqrt[3]{4 k}}^{2}}$

Step 3.3

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

$\frac{\sqrt[3]{3} {\sqrt[3]{4 k}}^{2}}{{\sqrt[3]{4 k}}^{1 + 2}}$

Step 3.4

Add $1$ and $2$ .

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

Step 3.5

Rewrite ${\sqrt[3]{4 k}}^{3}$ as $4 k$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 k}$ as ${(4 k)}^{\frac{1}{3}}$ .

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.1

Cancel the common factor.

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

Step 3.5.4.2

Rewrite the expression.

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

Step 3.5.5

Simplify.

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

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[3]{4 k}}^{2}$ as $\sqrt[3]{{(4 k)}^{2}}$ .

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

Step 4.2

Apply the product rule to $4 k$ .

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

Step 4.3

Raise $4$ to the power of $2$ .

$\frac{\sqrt[3]{3} \sqrt[3]{16 k^{2}}}{4 k}$

Step 4.4

Rewrite $16 k^{2}$ as $2^{3} \cdot (2 k^{2})$ .

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Step 4.4.1

Factor $8$ out of $16$ .

$\frac{\sqrt[3]{3} \sqrt[3]{8 (2) k^{2}}}{4 k}$

Step 4.4.2

Rewrite $8$ as $2^{3}$ .

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

Step 4.4.3

Add parentheses.

$\frac{\sqrt[3]{3} \sqrt[3]{2^{3} \cdot (2 k^{2})}}{4 k}$

Step 4.5

Pull terms out from under the radical.

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

Step 4.6

Combine exponents.

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Step 4.6.1

Combine using the product rule for radicals.

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

Step 4.6.2

Multiply $3$ by $2$ .

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

Step 5

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

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Step 5.1

Factor $2$ out of $2 \sqrt[3]{6 k^{2}}$ .

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $2$ out of $4 k$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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