Simplify square root of k^7 cube root of k^11

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

Step 1

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

Factor out $k^{6}$ .

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

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 ${(k^{3})}^{3} k^{2}$ .

Factor out $k^{9}$ .

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

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

Move $k^{3}$ .

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

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

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

Add $3$ and $3$ .

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

Step 6

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

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

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

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

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

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

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

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

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

$k^{6} (\sqrt[6]{k^{4}} k^{\frac{1}{2}})$

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

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

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

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

Combine using the product rule for radicals.

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

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

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

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

Add $4$ and $3$ .

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

Step 7

Factor out $k^{6}$ .

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

Step 8

Pull terms out from under the radical.

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

Step 9

Multiply $k^{6}$ by $k$ by adding the exponents.

Move $k$ .

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

Multiply $k$ by $k^{6}$ .

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

$k^{1} k^{6} \sqrt[6]{k}$

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

$k^{1 + 6} \sqrt[6]{k}$

Add $1$ and $6$ .

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

Simplify square root of k^7 cube root of k^11

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