Simplify fifth root of k^4 fourth root of k^11

$\sqrt[5]{k^{4}} \sqrt[4]{k^{11}}$

Rewrite $k^{11}$ as ${(k^{2})}^{4} k^{3}$ .

Factor out $k^{8}$ .

$\sqrt[5]{k^{4}} \sqrt[4]{k^{8} k^{3}}$

Rewrite $k^{8}$ as ${(k^{2})}^{4}$ .

$\sqrt[5]{k^{4}} \sqrt[4]{{(k^{2})}^{4} k^{3}}$

Pull terms out from under the radical.

$\sqrt[5]{k^{4}} (k^{2} \sqrt[4]{k^{3}})$

Multiply $\sqrt[5]{k^{4}} (k^{2} \sqrt[4]{k^{3}})$ .

Rewrite the expression using the least common index of $20$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{k^{4}}$ as $k^{\frac{4}{5}}$ .

$k^{2} (k^{\frac{4}{5}} \sqrt[4]{k^{3}})$

Rewrite $k^{\frac{4}{5}}$ as $k^{\frac{16}{20}}$ .

$k^{2} (k^{\frac{16}{20}} \sqrt[4]{k^{3}})$

Rewrite $k^{\frac{16}{20}}$ as $\sqrt[20]{k^{16}}$ .

$k^{2} (\sqrt[20]{k^{16}} \sqrt[4]{k^{3}})$

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

$k^{2} (\sqrt[20]{k^{16}} k^{\frac{3}{4}})$

Rewrite $k^{\frac{3}{4}}$ as $k^{\frac{15}{20}}$ .

$k^{2} (\sqrt[20]{k^{16}} k^{\frac{15}{20}})$

Rewrite $k^{\frac{15}{20}}$ as $\sqrt[20]{k^{15}}$ .

$k^{2} (\sqrt[20]{k^{16}} \sqrt[20]{k^{15}})$

Combine using the product rule for radicals.

$k^{2} \sqrt[20]{k^{16} k^{15}}$

Multiply $k^{16}$ by $k^{15}$ by adding the exponents.

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

$k^{2} \sqrt[20]{k^{16 + 15}}$

Add $16$ and $15$ .

$k^{2} \sqrt[20]{k^{31}}$

Factor out $k^{20}$ .

$k^{2} \sqrt[20]{k^{20} k^{11}}$

Pull terms out from under the radical.

$k^{2} (k \sqrt[20]{k^{11}})$

Multiply $k^{2}$ by $k$ by adding the exponents.

Move $k$ .

$k \cdot k^{2} \sqrt[20]{k^{11}}$

Multiply $k$ by $k^{2}$ .

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

$k^{1} k^{2} \sqrt[20]{k^{11}}$

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

$k^{1 + 2} \sqrt[20]{k^{11}}$

Add $1$ and $2$ .

$k^{3} \sqrt[20]{k^{11}}$