Algebra Examples

$\frac{\sqrt{a} + \sqrt{b}}{a^{2} - b^{2}}$

Step 1

Multiply to rationalize the numerator.

$\frac{(\sqrt{a} + \sqrt{b}) (\sqrt{a} - \sqrt{b})}{(a^{2} - b^{2}) (\sqrt{a} - \sqrt{b})}$

Step 2

Simplify.

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Step 2.1

Expand the numerator using the FOIL method.

$\frac{{\sqrt{a}}^{2} - \sqrt{a b} + \sqrt{a b} - {\sqrt{b}}^{2}}{(a^{2} - b^{2}) (\sqrt{a} - \sqrt{b})}$

Step 2.2

Simplify.

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Step 2.2.1

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

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Step 2.2.1.1

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

$\frac{{(a^{\frac{1}{2}})}^{2} - {\sqrt{b}}^{2}}{(a^{2} - b^{2}) (\sqrt{a} - \sqrt{b})}$

Step 2.2.1.2

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

$\frac{a^{\frac{1}{2} \cdot 2} - {\sqrt{b}}^{2}}{(a^{2} - b^{2}) (\sqrt{a} - \sqrt{b})}$

Step 2.2.1.3

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

$\frac{a^{\frac{2}{2}} - {\sqrt{b}}^{2}}{(a^{2} - b^{2}) (\sqrt{a} - \sqrt{b})}$

Step 2.2.1.4

Cancel the common factor of $2$ .

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Step 2.2.1.4.1

Cancel the common factor.

$\frac{a^{\frac{2}{2}} - {\sqrt{b}}^{2}}{(a^{2} - b^{2}) (\sqrt{a} - \sqrt{b})}$

Step 2.2.1.4.2

Rewrite the expression.

$\frac{a^{1} - {\sqrt{b}}^{2}}{(a^{2} - b^{2}) (\sqrt{a} - \sqrt{b})}$

Step 2.2.1.5

Simplify.

$\frac{a - {\sqrt{b}}^{2}}{(a^{2} - b^{2}) (\sqrt{a} - \sqrt{b})}$

Step 2.2.2

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

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Step 2.2.2.1

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

$\frac{a - {(b^{\frac{1}{2}})}^{2}}{(a^{2} - b^{2}) (\sqrt{a} - \sqrt{b})}$

Step 2.2.2.2

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

$\frac{a - b^{\frac{1}{2} \cdot 2}}{(a^{2} - b^{2}) (\sqrt{a} - \sqrt{b})}$

Step 2.2.2.3

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

$\frac{a - b^{\frac{2}{2}}}{(a^{2} - b^{2}) (\sqrt{a} - \sqrt{b})}$

Step 2.2.2.4

Cancel the common factor of $2$ .

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Step 2.2.2.4.1

Cancel the common factor.

$\frac{a - b^{\frac{2}{2}}}{(a^{2} - b^{2}) (\sqrt{a} - \sqrt{b})}$

Step 2.2.2.4.2

Rewrite the expression.

$\frac{a - b^{1}}{(a^{2} - b^{2}) (\sqrt{a} - \sqrt{b})}$

Step 2.2.2.5

Simplify.

$\frac{a - b}{(a^{2} - b^{2}) (\sqrt{a} - \sqrt{b})}$

Step 3

Expand $(a^{2} - b^{2}) (\sqrt{a} - \sqrt{b})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{a - b}{a^{2} (\sqrt{a} - \sqrt{b}) - b^{2} (\sqrt{a} - \sqrt{b})}$

Step 3.2

Apply the distributive property.

$\frac{a - b}{a^{2} \sqrt{a} + a^{2} (- \sqrt{b}) - b^{2} (\sqrt{a} - \sqrt{b})}$

Step 3.3

Apply the distributive property.

$\frac{a - b}{a^{2} \sqrt{a} + a^{2} (- \sqrt{b}) - b^{2} \sqrt{a} - b^{2} (- \sqrt{b})}$

Step 4

Simplify each term.

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Step 4.1

Rewrite using the commutative property of multiplication.

$\frac{a - b}{a^{2} \sqrt{a} - a^{2} \sqrt{b} - b^{2} \sqrt{a} - b^{2} (- \sqrt{b})}$

Step 4.2

Multiply $- b^{2} (- \sqrt{b})$ .

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

$\frac{a - b}{a^{2} \sqrt{a} - a^{2} \sqrt{b} - b^{2} \sqrt{a} + 1 b^{2} \sqrt{b}}$

Step 4.2.2

Multiply $b^{2}$ by $1$ .

$\frac{a - b}{a^{2} \sqrt{a} - a^{2} \sqrt{b} - b^{2} \sqrt{a} + b^{2} \sqrt{b}}$