Simplify fourth root of 7^5* square root of 7^3

$\sqrt[4]{7^{5}} \cdot \sqrt{7^{3}}$

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

$\sqrt[4]{16807} \cdot \sqrt{7^{3}}$

Rewrite $16807$ as $7^{4} \cdot 7$ .

Factor $2401$ out of $16807$ .

$\sqrt[4]{2401 (7)} \cdot \sqrt{7^{3}}$

Rewrite $2401$ as $7^{4}$ .

$\sqrt[4]{7^{4} \cdot 7} \cdot \sqrt{7^{3}}$

Pull terms out from under the radical.

$7 \sqrt[4]{7} \cdot \sqrt{7^{3}}$

Raise $7$ to the power of $3$ .

$7 \sqrt[4]{7} \cdot \sqrt{343}$

Rewrite $343$ as $7^{2} \cdot 7$ .

Factor $49$ out of $343$ .

$7 \sqrt[4]{7} \cdot \sqrt{49 (7)}$

Rewrite $49$ as $7^{2}$ .

$7 \sqrt[4]{7} \cdot \sqrt{7^{2} \cdot 7}$

Pull terms out from under the radical.

$7 \sqrt[4]{7} \cdot (7 \sqrt{7})$

Multiply $7 \sqrt[4]{7} (7 \sqrt{7})$ .

Multiply $7$ by $7$ .

$49 \sqrt[4]{7} \sqrt{7}$

Rewrite the expression using the least common index of $4$ .

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

$49 (7^{\frac{1}{2}} \sqrt[4]{7})$

Rewrite $7^{\frac{1}{2}}$ as $7^{\frac{2}{4}}$ .

$49 (7^{\frac{2}{4}} \sqrt[4]{7})$

Rewrite $7^{\frac{2}{4}}$ as $\sqrt[4]{7^{2}}$ .

$49 (\sqrt[4]{7^{2}} \sqrt[4]{7})$

Combine using the product rule for radicals.

$49 \sqrt[4]{7^{2} \cdot 7}$

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

Multiply $7^{2}$ by $7$ .

Raise $7$ to the power of $1$ .

$49 \sqrt[4]{7^{2} \cdot 7^{1}}$

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

$49 \sqrt[4]{7^{2 + 1}}$

Add $2$ and $1$ .

$49 \sqrt[4]{7^{3}}$

Raise $7$ to the power of $3$ .

$49 \sqrt[4]{343}$

The result can be shown in multiple forms.

Exact Form:

$49 \sqrt[4]{343}$

Decimal Form:

$210.87233646 \dots$

Simplify fourth root of 7^5* square root of 7^3

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