Simplify the Radical Expression square root of 7* cube root of 7

$\sqrt{7} \cdot \sqrt[3]{7}$

Step 1

Multiply $\sqrt{7} \sqrt[3]{7}$ .

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$7^{\frac{1}{2}} \sqrt[3]{7}$

Rewrite $7^{\frac{1}{2}}$ as $7^{\frac{3}{6}}$ .

$7^{\frac{3}{6}} \sqrt[3]{7}$

Rewrite $7^{\frac{3}{6}}$ as $\sqrt[6]{7^{3}}$ .

$\sqrt[6]{7^{3}} \sqrt[3]{7}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$\sqrt[6]{7^{3}} \cdot 7^{\frac{1}{3}}$

Rewrite $7^{\frac{1}{3}}$ as $7^{\frac{2}{6}}$ .

$\sqrt[6]{7^{3}} \cdot 7^{\frac{2}{6}}$

Rewrite $7^{\frac{2}{6}}$ as $\sqrt[6]{7^{2}}$ .

$\sqrt[6]{7^{3}} \sqrt[6]{7^{2}}$

Combine using the product rule for radicals.

$\sqrt[6]{7^{3} \cdot 7^{2}}$

Multiply $7^{3}$ by $7^{2}$ by adding the exponents.

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

$\sqrt[6]{7^{3 + 2}}$

Add $3$ and $2$ .

$\sqrt[6]{7^{5}}$

Step 2

Raise $7$ to the power of $5$ .

$\sqrt[6]{16807}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{16807}$

Decimal Form:

$5.06114018 \dots$