Algebra Examples

$\sqrt[3]{x^{2}} \cdot \sqrt[4]{x}$

Step 1

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

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Step 1.1

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

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

Step 1.2

Rewrite $x^{\frac{2}{3}}$ as $x^{\frac{8}{12}}$ .

$x^{\frac{8}{12}} \sqrt[4]{x}$

Step 1.3

Rewrite $x^{\frac{8}{12}}$ as $\sqrt[12]{x^{8}}$ .

$\sqrt[12]{x^{8}} \sqrt[4]{x}$

Step 1.4

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

$\sqrt[12]{x^{8}} x^{\frac{1}{4}}$

Step 1.5

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

$\sqrt[12]{x^{8}} x^{\frac{3}{12}}$

Step 1.6

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

$\sqrt[12]{x^{8}} \sqrt[12]{x^{3}}$

Step 2

Combine using the product rule for radicals.

$\sqrt[12]{x^{8} x^{3}}$

Step 3

Multiply $x^{8}$ by $x^{3}$ by adding the exponents.

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Step 3.1

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

$\sqrt[12]{x^{8 + 3}}$

Step 3.2

Add $8$ and $3$ .

$\sqrt[12]{x^{11}}$