Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

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

Step 5

Apply the distributive property.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Add $1$ and $1$ .

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

Step 7

Simplify terms.

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Step 7.1

Rewrite using the commutative property of multiplication.

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

Step 7.2

Simplify each term.

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Step 7.2.1

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

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Step 7.2.1.1

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

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

Step 7.2.1.2

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

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

Step 7.2.1.3

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

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

Step 7.2.1.4

Cancel the common factor of $2$ .

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Step 7.2.1.4.1

Cancel the common factor.

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

Step 7.2.1.4.2

Rewrite the expression.

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

Step 7.2.1.5

Simplify.

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

Step 7.2.2

Combine using the product rule for radicals.

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

Step 8

Simplify the numerator.

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Step 8.1

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

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

Step 8.2

Apply the product rule to $y x$ .

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

Step 8.3

Factor $x^{\frac{1}{2}}$ out of $x - y^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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Step 8.3.1

Move $y^{\frac{1}{2}}$ .

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

Step 8.3.2

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

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

Step 8.3.3

Factor $x^{\frac{1}{2}}$ out of $x^{1}$ .

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

Step 8.3.4

Factor $x^{\frac{1}{2}}$ out of $- x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

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

Step 8.3.5

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} x^{\frac{1}{2}} + x^{\frac{1}{2}} (- y^{\frac{1}{2}})$ .

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