Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.1.3

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

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

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Cancel the common factor.

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

Step 2.3.1.4.2

Rewrite the expression.

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

Step 2.3.1.5

Simplify.

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

Step 2.3.2

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

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.2.4

Cancel the common factor of $2$ .

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Step 2.3.2.4.1

Cancel the common factor.

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

Step 2.3.2.4.2

Rewrite the expression.

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

Step 2.3.2.5

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $1$ .

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

Step 4

Rewrite ${(\sqrt{x} + \sqrt{y})}^{2}$ as $(\sqrt{x} + \sqrt{y}) (\sqrt{x} + \sqrt{y})$ .

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

Step 5

Expand $(\sqrt{x} + \sqrt{y}) (\sqrt{x} + \sqrt{y})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $\sqrt{x} \sqrt{x}$ .

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Step 6.1.1.1

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

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

Step 6.1.1.2

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

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

Step 6.1.1.3

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

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

Step 6.1.1.4

Add $1$ and $1$ .

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

Step 6.1.2

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

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Step 6.1.2.1

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

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

Step 6.1.2.2

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

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

Step 6.1.2.3

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

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

Step 6.1.2.4

Cancel the common factor of $2$ .

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Step 6.1.2.4.1

Cancel the common factor.

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

Step 6.1.2.4.2

Rewrite the expression.

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

Step 6.1.2.5

Simplify.

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

Step 6.1.3

Combine using the product rule for radicals.

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

Step 6.1.4

Combine using the product rule for radicals.

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

Step 6.1.5

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

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Step 6.1.5.1

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

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

Step 6.1.5.2

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

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

Step 6.1.5.3

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

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

Step 6.1.5.4

Add $1$ and $1$ .

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

Step 6.1.6

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

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Step 6.1.6.1

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

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

Step 6.1.6.2

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

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

Step 6.1.6.3

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

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

Step 6.1.6.4

Cancel the common factor of $2$ .

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Step 6.1.6.4.1

Cancel the common factor.

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

Step 6.1.6.4.2

Rewrite the expression.

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

Step 6.1.6.5

Simplify.

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

Step 6.2

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

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Step 6.2.1

Reorder $y$ and $x$ .

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

Step 6.2.2

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

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