Simplify cube root of 3* fifth root of 3^6

$\sqrt[3]{3} \cdot \sqrt[5]{3^{6}}$

Raise $3$ to the power of $6$ .

$\sqrt[3]{3} \cdot \sqrt[5]{729}$

Rewrite $729$ as $3^{5} \cdot 3$ .

Factor $243$ out of $729$ .

$\sqrt[3]{3} \cdot \sqrt[5]{243 (3)}$

Rewrite $243$ as $3^{5}$ .

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

Pull terms out from under the radical.

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

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

Rewrite the expression using the least common index of $15$ .

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

$3 (3^{\frac{1}{3}} \sqrt[5]{3})$

Rewrite $3^{\frac{1}{3}}$ as $3^{\frac{5}{15}}$ .

$3 (3^{\frac{5}{15}} \sqrt[5]{3})$

Rewrite $3^{\frac{5}{15}}$ as $\sqrt[15]{3^{5}}$ .

$3 (\sqrt[15]{3^{5}} \sqrt[5]{3})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{3}$ as $3^{\frac{1}{5}}$ .

$3 (\sqrt[15]{3^{5}} \cdot 3^{\frac{1}{5}})$

Rewrite $3^{\frac{1}{5}}$ as $3^{\frac{3}{15}}$ .

$3 (\sqrt[15]{3^{5}} \cdot 3^{\frac{3}{15}})$

Rewrite $3^{\frac{3}{15}}$ as $\sqrt[15]{3^{3}}$ .

$3 (\sqrt[15]{3^{5}} \sqrt[15]{3^{3}})$

Combine using the product rule for radicals.

$3 \sqrt[15]{3^{5} \cdot 3^{3}}$

Multiply $3^{5}$ by $3^{3}$ by adding the exponents.

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

$3 \sqrt[15]{3^{5 + 3}}$

Add $5$ and $3$ .

$3 \sqrt[15]{3^{8}}$

Raise $3$ to the power of $8$ .

$3 \sqrt[15]{6561}$

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt[15]{6561}$

Decimal Form:

$5.38996473 \dots$