Evaluate square root of 3^5* cube root of 3

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

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

$\sqrt{243} \cdot \sqrt[3]{3}$

Rewrite $243$ as $9^{2} \cdot 3$ .

Factor $81$ out of $243$ .

$\sqrt{81 (3)} \cdot \sqrt[3]{3}$

Rewrite $81$ as $9^{2}$ .

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

Pull terms out from under the radical.

$9 \sqrt{3} \cdot \sqrt[3]{3}$

Multiply $9 \sqrt{3} \sqrt[3]{3}$ .

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

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

$9 (3^{\frac{1}{3}} \sqrt{3})$

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

$9 (3^{\frac{2}{6}} \sqrt{3})$

Rewrite $3^{\frac{2}{6}}$ as $\sqrt[6]{3^{2}}$ .

$9 (\sqrt[6]{3^{2}} \sqrt{3})$

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

$9 (\sqrt[6]{3^{2}} \cdot 3^{\frac{1}{2}})$

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

$9 (\sqrt[6]{3^{2}} \cdot 3^{\frac{3}{6}})$

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

$9 (\sqrt[6]{3^{2}} \sqrt[6]{3^{3}})$

Combine using the product rule for radicals.

$9 \sqrt[6]{3^{2} \cdot 3^{3}}$

Multiply $3^{2}$ by $3^{3}$ by adding the exponents.

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

$9 \sqrt[6]{3^{2 + 3}}$

Add $2$ and $3$ .

$9 \sqrt[6]{3^{5}}$

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

$9 \sqrt[6]{243}$

The result can be shown in multiple forms.

Exact Form:

$9 \sqrt[6]{243}$

Decimal Form:

$22.48244579 \dots$