Algebra Examples

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

Step 1

Multiply $\frac{1}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

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

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{1}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

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

Step 2.2

Raise $\sqrt{y}$ to the power of $1$ .

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

Step 2.3

Raise $\sqrt{y}$ to the power of $1$ .

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Simplify.

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

Step 3

Simplify each term.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Expand the denominator using the FOIL method.

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

Step 3.4

Simplify.

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Step 3.4.1

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

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Step 3.4.1.1

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

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

Step 3.4.1.2

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

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

Step 3.4.1.3

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

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

Step 3.4.1.4

Cancel the common factor of $2$ .

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Step 3.4.1.4.1

Cancel the common factor.

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

Step 3.4.1.4.2

Rewrite the expression.

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

Step 3.4.1.5

Simplify.

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

Step 3.4.2

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

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Step 3.4.2.1

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

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

Step 3.4.2.2

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

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

Step 3.4.2.3

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

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

Step 3.4.2.4

Cancel the common factor of $2$ .

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Step 3.4.2.4.1

Cancel the common factor.

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

Step 3.4.2.4.2

Rewrite the expression.

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

Step 3.4.2.5

Simplify.

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Expand the denominator using the FOIL method.

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

Step 3.8

Simplify.

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $- - \sqrt{y}$ .

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Step 5.2.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.2

Multiply $\sqrt{y}$ by $1$ .

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

Step 6

Simplify by adding terms.

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Step 6.1

Combine the opposite terms in $\sqrt{x} + \sqrt{y} - \sqrt{x} + \sqrt{y}$ .

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Step 6.1.1

Subtract $\sqrt{x}$ from $\sqrt{x}$ .

$\frac{\sqrt{y}}{y} \cdot \frac{0 + \sqrt{y} + \sqrt{y}}{x - y}$

Step 6.1.2

Add $0$ and $\sqrt{y}$ .

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

Step 6.2

Add $\sqrt{y}$ and $\sqrt{y}$ .

$\frac{\sqrt{y}}{y} \cdot \frac{2 \sqrt{y}}{x - y}$

Step 7

Multiply $\frac{\sqrt{y}}{y} \cdot \frac{2 \sqrt{y}}{x - y}$ .

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Step 7.1

Multiply $\frac{\sqrt{y}}{y}$ by $\frac{2 \sqrt{y}}{x - y}$ .

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

Step 7.2

Raise $\sqrt{y}$ to the power of $1$ .

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

Step 7.3

Raise $\sqrt{y}$ to the power of $1$ .

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Cancel the common factor of $2$ .

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Step 8.4.1

Cancel the common factor.

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

Step 8.4.2

Rewrite the expression.

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

Step 8.5

Simplify.

$\frac{2 y}{y (x - y)}$

Step 9

Cancel the common factor of $y$ .

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Step 9.1

Cancel the common factor.

$\frac{2 y}{y (x - y)}$

Step 9.2

Rewrite the expression.

$\frac{2}{x - y}$