Evaluate square root of 2* fifth root of 5

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

Step 1

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

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

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

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

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

Rewrite $2^{\frac{5}{10}}$ as $\sqrt[10]{2^{5}}$ .

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

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

$\sqrt[10]{2^{5}} \cdot 5^{\frac{1}{5}}$

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

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

Rewrite $5^{\frac{2}{10}}$ as $\sqrt[10]{5^{2}}$ .

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

Step 2

Combine using the product rule for radicals.

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

Step 3

Raise $2$ to the power of $5$ .

$\sqrt[10]{32 \cdot 5^{2}}$

Step 4

Raise $5$ to the power of $2$ .

$\sqrt[10]{32 \cdot 25}$

Step 5

Multiply $32$ by $25$ .

$\sqrt[10]{800}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt[10]{800}$

Decimal Form:

$1.95123239 \dots$