Simplify square root of 3* fifth root of 3^2

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

Step 1

Raise $3$ to the power of $2$ .

$\sqrt{3} \cdot \sqrt[5]{9}$

Step 2

Multiply $\sqrt{3} \sqrt[5]{9}$ .

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

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

$3^{\frac{1}{2}} \sqrt[5]{9}$

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

$3^{\frac{5}{10}} \sqrt[5]{9}$

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

$\sqrt[10]{3^{5}} \sqrt[5]{9}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{9}$ as $9^{\frac{1}{5}}$ .

$\sqrt[10]{3^{5}} \cdot 9^{\frac{1}{5}}$

Rewrite $9^{\frac{1}{5}}$ as $9^{\frac{2}{10}}$ .

$\sqrt[10]{3^{5}} \cdot 9^{\frac{2}{10}}$

Rewrite $9^{\frac{2}{10}}$ as $\sqrt[10]{9^{2}}$ .

$\sqrt[10]{3^{5}} \sqrt[10]{9^{2}}$

Combine using the product rule for radicals.

$\sqrt[10]{3^{5} \cdot 9^{2}}$

Rewrite $9$ as $3^{2}$ .

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

Multiply the exponents in ${(3^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $2$ by $2$ .

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

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

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

Add $5$ and $4$ .

$\sqrt[10]{3^{9}}$

Step 3

Raise $3$ to the power of $9$ .

$\sqrt[10]{19683}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt[10]{19683}$

Decimal Form:

$2.68787537 \dots$