Evaluate square root of 2* cube root of 5

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

Step 1

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

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

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

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

$2^{\frac{3}{6}} \sqrt[3]{5}$

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

$\sqrt[6]{2^{3}} \sqrt[3]{5}$

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

$\sqrt[6]{2^{3}} \cdot 5^{\frac{1}{3}}$

Rewrite $5^{\frac{1}{3}}$ as $5^{\frac{2}{6}}$ .

$\sqrt[6]{2^{3}} \cdot 5^{\frac{2}{6}}$

Rewrite $5^{\frac{2}{6}}$ as $\sqrt[6]{5^{2}}$ .

$\sqrt[6]{2^{3}} \sqrt[6]{5^{2}}$

Step 2

Combine using the product rule for radicals.

$\sqrt[6]{2^{3} \cdot 5^{2}}$

Step 3

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

$\sqrt[6]{8 \cdot 5^{2}}$

Step 4

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

$\sqrt[6]{8 \cdot 25}$

Step 5

Multiply $8$ by $25$ .

$\sqrt[6]{200}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{200}$

Decimal Form:

$2.41827117 \dots$