Algebra Examples

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

Step 1

Factor out $a^{2}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

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

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

Step 3.1.6

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

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

Step 3.2

Combine using the product rule for radicals.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Add $2$ and $3$ .

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