Find the Product square root of 5* cube root of 5

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

Step 1

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

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

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

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

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

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

Rewrite $5^{\frac{3}{6}}$ as $\sqrt[6]{5^{3}}$ .

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

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

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

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

Add $3$ and $2$ .

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

Step 2

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

$\sqrt[6]{3125}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{3125}$

Decimal Form:

$3.82362245 \dots$