Algebra Examples

$\sqrt{k^{7}} \sqrt[3]{k^{11}}$

Step 1

Rewrite $k^{7}$ as ${(k^{3})}^{2} k$ .

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Step 1.1

Factor out $k^{6}$ .

$\sqrt{k^{6} k} \sqrt[3]{k^{11}}$

Step 1.2

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

$\sqrt{{(k^{3})}^{2} k} \sqrt[3]{k^{11}}$

Step 2

Pull terms out from under the radical.

$| k^{3} | \sqrt{k} \sqrt[3]{k^{11}}$

Step 3

Rewrite $k^{11}$ as ${(k^{3})}^{3} k^{2}$ .

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Step 3.1

Factor out $k^{9}$ .

$| k^{3} | \sqrt{k} \sqrt[3]{k^{9} k^{2}}$

Step 3.2

Rewrite $k^{9}$ as ${(k^{3})}^{3}$ .

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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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 $k^{\frac{1}{2}}$ .

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

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

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

Step 5.1.5

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

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

Step 5.1.6

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

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

Step 5.2

Combine using the product rule for radicals.

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

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.

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

Step 5.3.2

Add $3$ and $4$ .

$| k^{3} | (k^{3} \sqrt[6]{k^{7}})$

Step 6

Factor out $k^{6}$ .

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

Step 7

Pull terms out from under the radical.

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

Step 8

Multiply $| k^{3} | (k^{3} (| k | \sqrt[6]{k}))$ .

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Step 8.1

To multiply absolute values, multiply the terms inside each absolute value.

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

Step 8.2

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

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Step 8.2.1

Multiply $k^{3}$ by $k$ .

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Step 8.2.1.1

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

$k^{3} (| k^{3} k^{1} | \sqrt[6]{k})$

Step 8.2.1.2

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

$k^{3} (| k^{3 + 1} | \sqrt[6]{k})$

Step 8.2.2

Add $3$ and $1$ .

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

Step 9

Remove non-negative terms from the absolute value.

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

Step 10

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

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Step 10.1

Move $k^{4}$ .

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

Step 10.2

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

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

Step 10.3

Add $4$ and $3$ .

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