Simplify ( square root of a^5)/( cube root of a^2)

$\frac{\sqrt{a^{5}}}{\sqrt[3]{a^{2}}}$

Step 1

Simplify the numerator.

Rewrite $a^{5}$ as ${(a^{2})}^{2} a$ .

Factor out $a^{4}$ .

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

Rewrite $a^{4}$ as ${(a^{2})}^{2}$ .

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

Pull terms out from under the radical.

$\frac{a^{2} \sqrt{a}}{\sqrt[3]{a^{2}}}$

Step 2

Multiply $\frac{a^{2} \sqrt{a}}{\sqrt[3]{a^{2}}}$ by $\frac{{\sqrt[3]{a^{2}}}^{2}}{{\sqrt[3]{a^{2}}}^{2}}$ .

$\frac{a^{2} \sqrt{a}}{\sqrt[3]{a^{2}}} \cdot \frac{{\sqrt[3]{a^{2}}}^{2}}{{\sqrt[3]{a^{2}}}^{2}}$

Step 3

Combine and simplify the denominator.

Multiply $\frac{a^{2} \sqrt{a}}{\sqrt[3]{a^{2}}}$ by $\frac{{\sqrt[3]{a^{2}}}^{2}}{{\sqrt[3]{a^{2}}}^{2}}$ .

$\frac{a^{2} \sqrt{a} {\sqrt[3]{a^{2}}}^{2}}{\sqrt[3]{a^{2}} {\sqrt[3]{a^{2}}}^{2}}$

Raise $\sqrt[3]{a^{2}}$ to the power of $1$ .

$\frac{a^{2} \sqrt{a} {\sqrt[3]{a^{2}}}^{2}}{{\sqrt[3]{a^{2}}}^{1} {\sqrt[3]{a^{2}}}^{2}}$

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

$\frac{a^{2} \sqrt{a} {\sqrt[3]{a^{2}}}^{2}}{{\sqrt[3]{a^{2}}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{a^{2} \sqrt{a} {\sqrt[3]{a^{2}}}^{2}}{{\sqrt[3]{a^{2}}}^{3}}$

Rewrite ${\sqrt[3]{a^{2}}}^{3}$ as $a^{2}$ .

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

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{a^{2} \sqrt{a} {\sqrt[3]{a^{2}}}^{2}}{a^{\frac{2}{3} \cdot 3}}$

Combine $\frac{2}{3}$ and $3$ .

$\frac{a^{2} \sqrt{a} {\sqrt[3]{a^{2}}}^{2}}{a^{\frac{2 \cdot 3}{3}}}$

Multiply $2$ by $3$ .

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

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6$ .

$\frac{a^{2} \sqrt{a} {\sqrt[3]{a^{2}}}^{2}}{a^{\frac{3 \cdot 2}{3}}}$

Cancel the common factors.

Factor $3$ out of $3$ .

$\frac{a^{2} \sqrt{a} {\sqrt[3]{a^{2}}}^{2}}{a^{\frac{3 \cdot 2}{3 (1)}}}$

Cancel the common factor.

$\frac{a^{2} \sqrt{a} {\sqrt[3]{a^{2}}}^{2}}{a^{\frac{3 \cdot 2}{3 \cdot 1}}}$

Rewrite the expression.

$\frac{a^{2} \sqrt{a} {\sqrt[3]{a^{2}}}^{2}}{a^{\frac{2}{1}}}$

Divide $2$ by $1$ .

$\frac{a^{2} \sqrt{a} {\sqrt[3]{a^{2}}}^{2}}{a^{2}}$

Step 4

Cancel the common factor of $a^{2}$ .

Cancel the common factor.

$\frac{a^{2} \sqrt{a} {\sqrt[3]{a^{2}}}^{2}}{a^{2}}$

Divide $\sqrt{a} {\sqrt[3]{a^{2}}}^{2}$ by $1$ .

$\sqrt{a} {\sqrt[3]{a^{2}}}^{2}$

Step 5

Rewrite ${\sqrt[3]{a^{2}}}^{2}$ as ${({(a^{2})}^{2})}^{\frac{1}{3}}$ .

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

Step 6

Multiply the exponents in ${(a^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{a} \sqrt[3]{a^{2 \cdot 2}}$

Multiply $2$ by $2$ .

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

Step 7

Factor out $a^{3}$ .

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

Step 8

Pull terms out from under the radical.

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

Step 9

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

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

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

$a (a^{\frac{1}{2}} \sqrt[3]{a})$

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

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

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

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

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

$a (\sqrt[6]{a^{3}} a^{\frac{1}{3}})$

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

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

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

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

Combine using the product rule for radicals.

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

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

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

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

Add $3$ and $2$ .

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

Simplify ( square root of a^5)/( cube root of a^2)

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