Algebra Examples

$\frac{\sqrt[3]{k^{8}}}{\sqrt{k}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $k^{8}$ as ${(k^{2})}^{3} k^{2}$ .

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Step 1.1.1

Factor out $k^{6}$ .

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

Step 1.1.2

Rewrite $k^{6}$ as ${(k^{2})}^{3}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

Raise $\sqrt{k}$ to the power of $1$ .

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

Step 3.3

Raise $\sqrt{k}$ to the power of $1$ .

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Rewrite ${\sqrt{k}}^{2}$ as $k$ .

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

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

Step 3.6.4.2

Rewrite the expression.

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

Step 3.6.5

Simplify.

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

Step 4

Cancel the common factor of $k^{2}$ and $k$ .

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Step 4.1

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

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Raise $k$ to the power of $1$ .

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

Step 4.2.2

Factor $k$ out of $k^{1}$ .

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

Step 4.2.3

Cancel the common factor.

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

Step 4.2.4

Rewrite the expression.

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

Step 4.2.5

Divide $k \sqrt[3]{k^{2}} \sqrt{k}$ by $1$ .

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

Step 5

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

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Step 5.1

Rewrite the expression using the least common index of $6$ .

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Step 5.1.1

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

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

Step 5.1.2

Rewrite $k^{\frac{1}{2}}$ as $k^{\frac{3}{6}}$ .

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

Step 5.1.3

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

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

Step 5.1.4

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

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

Step 5.1.5

Rewrite $k^{\frac{2}{3}}$ as $k^{\frac{4}{6}}$ .

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

Step 5.1.6

Rewrite $k^{\frac{4}{6}}$ as $\sqrt[6]{k^{4}}$ .

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

Step 5.2

Combine using the product rule for radicals.

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

Step 5.3

Multiply $k^{3}$ by $k^{4}$ by adding the exponents.

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Step 5.3.1

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

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

Step 5.3.2

Add $3$ and $4$ .

$k \sqrt[6]{k^{7}}$

Step 6

Factor out $k^{6}$ .

$k \sqrt[6]{k^{6} k}$

Step 7

Pull terms out from under the radical.

$k (k \sqrt[6]{k})$

Step 8

Multiply $k$ by $k$ by adding the exponents.

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Step 8.1

Move $k$ .

$k \cdot k \sqrt[6]{k}$

Step 8.2

Multiply $k$ by $k$ .

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