Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite the expression using the least common index of $4$ .

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Step 1.1.1

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

$\frac{{(x y)}^{\frac{1}{2}} \sqrt[4]{x^{2} y}}{x y}$

Step 1.1.2

Rewrite ${(x y)}^{\frac{1}{2}}$ as ${(x y)}^{\frac{2}{4}}$ .

$\frac{{(x y)}^{\frac{2}{4}} \sqrt[4]{x^{2} y}}{x y}$

Step 1.1.3

Rewrite ${(x y)}^{\frac{2}{4}}$ as $\sqrt[4]{{(x y)}^{2}}$ .

$\frac{\sqrt[4]{{(x y)}^{2}} \sqrt[4]{x^{2} y}}{x y}$

Step 1.2

Combine using the product rule for radicals.

$\frac{\sqrt[4]{{(x y)}^{2} (x^{2} y)}}{x y}$

Step 1.3

Apply the product rule to $x y$ .

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

Step 1.4

Combine exponents.

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Step 1.4.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.4.1.1

Move $x^{2}$ .

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

Step 1.4.1.2

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

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

Step 1.4.1.3

Add $2$ and $2$ .

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

Step 1.4.2

Raise $y$ to the power of $1$ .

$\frac{\sqrt[4]{x^{4} (y^{1} y^{2})}}{x y}$

Step 1.4.3

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

$\frac{\sqrt[4]{x^{4} y^{1 + 2}}}{x y}$

Step 1.4.4

Add $1$ and $2$ .

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

Step 2

Simplify terms.

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Step 2.1

Pull terms out from under the radical.

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

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

$\frac{\sqrt[4]{y^{3}}}{y}$