Simplify the Radical Expression square root of 7* fifth root of 7^6

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

Raise $7$ to the power of $6$ .

$\sqrt{7} \cdot \sqrt[5]{117649}$

Rewrite $117649$ as $7^{5} \cdot 7$ .

Factor $16807$ out of $117649$ .

$\sqrt{7} \cdot \sqrt[5]{16807 (7)}$

Rewrite $16807$ as $7^{5}$ .

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

Pull terms out from under the radical.

$\sqrt{7} \cdot (7 \sqrt[5]{7})$

Multiply $\sqrt{7} (7 \sqrt[5]{7})$ .

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

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

$7 (7^{\frac{1}{2}} \sqrt[5]{7})$

Rewrite $7^{\frac{1}{2}}$ as $7^{\frac{5}{10}}$ .

$7 (7^{\frac{5}{10}} \sqrt[5]{7})$

Rewrite $7^{\frac{5}{10}}$ as $\sqrt[10]{7^{5}}$ .

$7 (\sqrt[10]{7^{5}} \sqrt[5]{7})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{7}$ as $7^{\frac{1}{5}}$ .

$7 (\sqrt[10]{7^{5}} \cdot 7^{\frac{1}{5}})$

Rewrite $7^{\frac{1}{5}}$ as $7^{\frac{2}{10}}$ .

$7 (\sqrt[10]{7^{5}} \cdot 7^{\frac{2}{10}})$

Rewrite $7^{\frac{2}{10}}$ as $\sqrt[10]{7^{2}}$ .

$7 (\sqrt[10]{7^{5}} \sqrt[10]{7^{2}})$

Combine using the product rule for radicals.

$7 \sqrt[10]{7^{5} \cdot 7^{2}}$

Multiply $7^{5}$ by $7^{2}$ by adding the exponents.

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

$7 \sqrt[10]{7^{5 + 2}}$

Add $5$ and $2$ .

$7 \sqrt[10]{7^{7}}$

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

$7 \sqrt[10]{823543}$

The result can be shown in multiple forms.

Exact Form:

$7 \sqrt[10]{823543}$

Decimal Form:

$27.33170143 \dots$