Evaluate ( square root of 3)( fifth root of 3)

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

Step 1

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

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

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

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

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

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

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

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

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

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

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

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

Rewrite $3^{\frac{2}{10}}$ as $\sqrt[10]{3^{2}}$ .

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

Combine using the product rule for radicals.

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

Multiply $3^{5}$ by $3^{2}$ by adding the exponents.

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

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

Add $5$ and $2$ .

$\sqrt[10]{3^{7}}$

Step 2

Raise $3$ to the power of $7$ .

$\sqrt[10]{2187}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt[10]{2187}$

Decimal Form:

$2.15766927 \dots$