Simplify square root of 12* fifth root of 12^3

$\sqrt{12} \cdot \sqrt[5]{12^{3}}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$\sqrt{4 (3)} \cdot \sqrt[5]{12^{3}}$

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 3} \cdot \sqrt[5]{12^{3}}$

Pull terms out from under the radical.

$2 \sqrt{3} \cdot \sqrt[5]{12^{3}}$

Raise $12$ to the power of $3$ .

$2 \sqrt{3} \cdot \sqrt[5]{1728}$

Rewrite $1728$ as $2^{5} \cdot 54$ .

Factor $32$ out of $1728$ .

$2 \sqrt{3} \cdot \sqrt[5]{32 (54)}$

Rewrite $32$ as $2^{5}$ .

$2 \sqrt{3} \cdot \sqrt[5]{2^{5} \cdot 54}$

Pull terms out from under the radical.

$2 \sqrt{3} \cdot (2 \sqrt[5]{54})$

Multiply $2 \sqrt{3} (2 \sqrt[5]{54})$ .

Multiply $2$ by $2$ .

$4 \sqrt{3} \sqrt[5]{54}$

Rewrite the expression using the least common index of $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{54}$ as $54^{\frac{1}{5}}$ .

$4 (54^{\frac{1}{5}} \sqrt{3})$

Rewrite $54^{\frac{1}{5}}$ as $54^{\frac{2}{10}}$ .

$4 (54^{\frac{2}{10}} \sqrt{3})$

Rewrite $54^{\frac{2}{10}}$ as $\sqrt[10]{54^{2}}$ .

$4 (\sqrt[10]{54^{2}} \sqrt{3})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$4 (\sqrt[10]{54^{2}} \cdot 3^{\frac{1}{2}})$

Rewrite $3^{\frac{1}{2}}$ as $3^{\frac{5}{10}}$ .

$4 (\sqrt[10]{54^{2}} \cdot 3^{\frac{5}{10}})$

Rewrite $3^{\frac{5}{10}}$ as $\sqrt[10]{3^{5}}$ .

$4 (\sqrt[10]{54^{2}} \sqrt[10]{3^{5}})$

Combine using the product rule for radicals.

$4 \sqrt[10]{54^{2} \cdot 3^{5}}$

Raise $54$ to the power of $2$ .

$4 \sqrt[10]{2916 \cdot 3^{5}}$

Raise $3$ to the power of $5$ .

$4 \sqrt[10]{2916 \cdot 243}$

Multiply $2916$ by $243$ .

$4 \sqrt[10]{708588}$

Rewrite $708588$ as $3^{10} \cdot 12$ .

Factor $59049$ out of $708588$ .

$4 \sqrt[10]{59049 (12)}$

Rewrite $59049$ as $3^{10}$ .

$4 \sqrt[10]{3^{10} \cdot 12}$

Pull terms out from under the radical.

$4 (3 \sqrt[10]{12})$

Multiply $3$ by $4$ .

$12 \sqrt[10]{12}$

The result can be shown in multiple forms.

Exact Form:

$12 \sqrt[10]{12}$

Decimal Form:

$15.38506624 \dots$

Simplify square root of 12* fifth root of 12^3

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