Algebra Examples

$1 \div (\sqrt{x} - \sqrt{y})$

Step 1

Rewrite the division as a fraction.

$\frac{1}{\sqrt{x} - \sqrt{y}}$

Step 2

Multiply $\frac{1}{\sqrt{x} - \sqrt{y}}$ by $\frac{\sqrt{x} + \sqrt{y}}{\sqrt{x} + \sqrt{y}}$ .

$\frac{1}{\sqrt{x} - \sqrt{y}} \cdot \frac{\sqrt{x} + \sqrt{y}}{\sqrt{x} + \sqrt{y}}$

Step 3

Multiply $\frac{1}{\sqrt{x} - \sqrt{y}}$ by $\frac{\sqrt{x} + \sqrt{y}}{\sqrt{x} + \sqrt{y}}$ .

$\frac{\sqrt{x} + \sqrt{y}}{(\sqrt{x} - \sqrt{y}) (\sqrt{x} + \sqrt{y})}$

Step 4

Expand the denominator using the FOIL method.

$\frac{\sqrt{x} + \sqrt{y}}{{\sqrt{x}}^{2} + \sqrt{x y} - \sqrt{x y} - {\sqrt{y}}^{2}}$

Step 5

Simplify.

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Step 5.1

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 5.1.1

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

$\frac{\sqrt{x} + \sqrt{y}}{{(x^{\frac{1}{2}})}^{2} - {\sqrt{y}}^{2}}$

Step 5.1.2

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

$\frac{\sqrt{x} + \sqrt{y}}{x^{\frac{1}{2} \cdot 2} - {\sqrt{y}}^{2}}$

Step 5.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{x} + \sqrt{y}}{x^{\frac{2}{2}} - {\sqrt{y}}^{2}}$

Step 5.1.4

Cancel the common factor of $2$ .

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Step 5.1.4.1

Cancel the common factor.

$\frac{\sqrt{x} + \sqrt{y}}{x^{\frac{2}{2}} - {\sqrt{y}}^{2}}$

Step 5.1.4.2

Rewrite the expression.

$\frac{\sqrt{x} + \sqrt{y}}{x^{1} - {\sqrt{y}}^{2}}$

Step 5.1.5

Simplify.

$\frac{\sqrt{x} + \sqrt{y}}{x - {\sqrt{y}}^{2}}$

Step 5.2

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

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Step 5.2.1

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

$\frac{\sqrt{x} + \sqrt{y}}{x - {(y^{\frac{1}{2}})}^{2}}$

Step 5.2.2

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

$\frac{\sqrt{x} + \sqrt{y}}{x - y^{\frac{1}{2} \cdot 2}}$

Step 5.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{x} + \sqrt{y}}{x - y^{\frac{2}{2}}}$

Step 5.2.4

Cancel the common factor of $2$ .

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Step 5.2.4.1

Cancel the common factor.

$\frac{\sqrt{x} + \sqrt{y}}{x - y^{\frac{2}{2}}}$

Step 5.2.4.2

Rewrite the expression.

$\frac{\sqrt{x} + \sqrt{y}}{x - y^{1}}$

Step 5.2.5

Simplify.

$\frac{\sqrt{x} + \sqrt{y}}{x - y}$