Evaluate square root of 5* fifth root of 3

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

Step 1

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

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

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

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

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

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

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

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

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

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

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

Step 2

Combine using the product rule for radicals.

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

Step 3

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

$\sqrt[10]{3125 \cdot 3^{2}}$

Step 4

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

$\sqrt[10]{3125 \cdot 9}$

Step 5

Multiply $3125$ by $9$ .

$\sqrt[10]{28125}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt[10]{28125}$

Decimal Form:

$2.78553906 \dots$