Simplify 5 cube root of a*4 square root of a^3

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

Factor out $a^{2}$ .

$5 \sqrt[3]{a} \cdot (4 \sqrt{a^{2} a})$

Pull terms out from under the radical.

$5 \sqrt[3]{a} \cdot (4 (a \sqrt{a}))$

Multiply $5 \sqrt[3]{a} (4 a \sqrt{a})$ .

Multiply $4$ by $5$ .

$20 \sqrt[3]{a} (a \sqrt{a})$

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

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

$20 (a (a^{\frac{1}{3}} \sqrt{a}))$

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

$20 (a (a^{\frac{2}{6}} \sqrt{a}))$

Rewrite $a^{\frac{2}{6}}$ as $\sqrt[6]{a^{2}}$ .

$20 (a (\sqrt[6]{a^{2}} \sqrt{a}))$

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

$20 (a (\sqrt[6]{a^{2}} a^{\frac{1}{2}}))$

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

$20 (a (\sqrt[6]{a^{2}} a^{\frac{3}{6}}))$

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

$20 (a (\sqrt[6]{a^{2}} \sqrt[6]{a^{3}}))$

Combine using the product rule for radicals.

$20 (a \sqrt[6]{a^{2} a^{3}})$

Multiply $a^{2}$ by $a^{3}$ by adding the exponents.

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

$20 (a \sqrt[6]{a^{2 + 3}})$

Add $2$ and $3$ .

$20 (a \sqrt[6]{a^{5}})$

$20 a \sqrt[6]{a^{5}}$