Simplify cube root of 5* square root of 2

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

Step 1

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

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

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

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

$5^{\frac{2}{6}} \sqrt{2}$

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

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

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

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

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

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

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

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

Step 2

Combine using the product rule for radicals.

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

Step 3

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

$\sqrt[6]{25 \cdot 2^{3}}$

Step 4

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

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

Step 5

Multiply $25$ by $8$ .

$\sqrt[6]{200}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{200}$

Decimal Form:

$2.41827117 \dots$