Simplify (4 square root of a)( cube root of a^2)

$(4 \sqrt{a}) (\sqrt[3]{a^{2}})$

Multiply $(4 \sqrt{a}) (\sqrt[3]{a^{2}})$ .

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{a^{2}}$ as $a^{\frac{2}{3}}$ .

$4 (a^{\frac{2}{3}} \sqrt{a})$

Rewrite $a^{\frac{2}{3}}$ as $a^{\frac{4}{6}}$ .

$4 (a^{\frac{4}{6}} \sqrt{a})$

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

$4 (\sqrt[6]{a^{4}} \sqrt{a})$

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

$4 (\sqrt[6]{a^{4}} a^{\frac{1}{2}})$

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

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

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

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

Combine using the product rule for radicals.

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

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

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

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

Add $4$ and $3$ .

$4 \sqrt[6]{a^{7}}$

Factor out $a^{6}$ .

$4 \sqrt[6]{a^{6} a}$

Pull terms out from under the radical.

$4 a \sqrt[6]{a}$