Simplify the Radical Expression ( fourth root of 6)( cube root of 6)

$(\sqrt[4]{6}) (\sqrt[3]{6})$

Multiply $(\sqrt[4]{6}) (\sqrt[3]{6})$ .

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{6}$ as $6^{\frac{1}{4}}$ .

$6^{\frac{1}{4}} \sqrt[3]{6}$

Rewrite $6^{\frac{1}{4}}$ as $6^{\frac{3}{12}}$ .

$6^{\frac{3}{12}} \sqrt[3]{6}$

Rewrite $6^{\frac{3}{12}}$ as $\sqrt[12]{6^{3}}$ .

$\sqrt[12]{6^{3}} \sqrt[3]{6}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6}$ as $6^{\frac{1}{3}}$ .

$\sqrt[12]{6^{3}} \cdot 6^{\frac{1}{3}}$

Rewrite $6^{\frac{1}{3}}$ as $6^{\frac{4}{12}}$ .

$\sqrt[12]{6^{3}} \cdot 6^{\frac{4}{12}}$

Rewrite $6^{\frac{4}{12}}$ as $\sqrt[12]{6^{4}}$ .

$\sqrt[12]{6^{3}} \sqrt[12]{6^{4}}$

Combine using the product rule for radicals.

$\sqrt[12]{6^{3} \cdot 6^{4}}$

Multiply $6^{3}$ by $6^{4}$ by adding the exponents.

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

$\sqrt[12]{6^{3 + 4}}$

Add $3$ and $4$ .

$\sqrt[12]{6^{7}}$

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

$\sqrt[12]{279936}$

The result can be shown in multiple forms.

Exact Form:

$\sqrt[12]{279936}$

Decimal Form:

$2.84394741 \dots$