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

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

Step 1

Multiply $\sqrt[3]{7} \sqrt[4]{7}$ .

Rewrite the expression using the least common index of $12$ .

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

$7^{\frac{1}{3}} \sqrt[4]{7}$

Rewrite $7^{\frac{1}{3}}$ as $7^{\frac{4}{12}}$ .

$7^{\frac{4}{12}} \sqrt[4]{7}$

Rewrite $7^{\frac{4}{12}}$ as $\sqrt[12]{7^{4}}$ .

$\sqrt[12]{7^{4}} \sqrt[4]{7}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{7}$ as $7^{\frac{1}{4}}$ .

$\sqrt[12]{7^{4}} \cdot 7^{\frac{1}{4}}$

Rewrite $7^{\frac{1}{4}}$ as $7^{\frac{3}{12}}$ .

$\sqrt[12]{7^{4}} \cdot 7^{\frac{3}{12}}$

Rewrite $7^{\frac{3}{12}}$ as $\sqrt[12]{7^{3}}$ .

$\sqrt[12]{7^{4}} \sqrt[12]{7^{3}}$

Combine using the product rule for radicals.

$\sqrt[12]{7^{4} \cdot 7^{3}}$

Multiply $7^{4}$ by $7^{3}$ by adding the exponents.

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

$\sqrt[12]{7^{4 + 3}}$

Add $4$ and $3$ .

$\sqrt[12]{7^{7}}$

Step 2

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

$\sqrt[12]{823543}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt[12]{823543}$

Decimal Form:

$3.11152902 \dots$