Algebra Examples

$\sqrt[4]{x^{2}} \cdot \sqrt[5]{x^{3}}$

Step 1

Rewrite $\sqrt[4]{x^{2}}$ as $\sqrt{\sqrt{x^{2}}}$ .

$\sqrt{\sqrt{x^{2}}} \cdot \sqrt[5]{x^{3}}$

Step 2

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt{x} \cdot \sqrt[5]{x^{3}}$

Step 3

Multiply $\sqrt{x} \sqrt[5]{x^{3}}$ .

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Step 3.1

Rewrite the expression using the least common index of $10$ .

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Step 3.1.1

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

$x^{\frac{1}{2}} \sqrt[5]{x^{3}}$

Step 3.1.2

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{5}{10}}$ .

$x^{\frac{5}{10}} \sqrt[5]{x^{3}}$

Step 3.1.3

Rewrite $x^{\frac{5}{10}}$ as $\sqrt[10]{x^{5}}$ .

$\sqrt[10]{x^{5}} \sqrt[5]{x^{3}}$

Step 3.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{3}}$ as $x^{\frac{3}{5}}$ .

$\sqrt[10]{x^{5}} x^{\frac{3}{5}}$

Step 3.1.5

Rewrite $x^{\frac{3}{5}}$ as $x^{\frac{6}{10}}$ .

$\sqrt[10]{x^{5}} x^{\frac{6}{10}}$

Step 3.1.6

Rewrite $x^{\frac{6}{10}}$ as $\sqrt[10]{x^{6}}$ .

$\sqrt[10]{x^{5}} \sqrt[10]{x^{6}}$

Step 3.2

Combine using the product rule for radicals.

$\sqrt[10]{x^{5} x^{6}}$

Step 3.3

Multiply $x^{5}$ by $x^{6}$ 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.

$\sqrt[10]{x^{5 + 6}}$

Step 3.3.2

Add $5$ and $6$ .

$\sqrt[10]{x^{11}}$

Step 4

Factor out $x^{10}$ .

$\sqrt[10]{x^{10} x}$

Step 5

Pull terms out from under the radical.

$x \sqrt[10]{x}$