Algebra Examples

$\frac{\sqrt{a b} - a}{\sqrt{b} - \sqrt{a}}$

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Apply the product rule to $a b$ .

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

Step 1.3

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} b^{\frac{1}{2}} - a$ .

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 2

Multiply $\frac{a^{\frac{1}{2}} (b^{\frac{1}{2}} - a^{\frac{1}{2}})}{\sqrt{b} - \sqrt{a}}$ by $\frac{\sqrt{b} + \sqrt{a}}{\sqrt{b} + \sqrt{a}}$ .

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

Step 3

Multiply $\frac{a^{\frac{1}{2}} (b^{\frac{1}{2}} - a^{\frac{1}{2}})}{\sqrt{b} - \sqrt{a}}$ by $\frac{\sqrt{b} + \sqrt{a}}{\sqrt{b} + \sqrt{a}}$ .

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

Step 4

Expand the denominator using the FOIL method.

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

Step 5

Simplify.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Cancel the common factor of $2$ .

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Step 5.1.4.1

Cancel the common factor.

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

Step 5.1.4.2

Rewrite the expression.

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

Step 5.1.5

Simplify.

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

Cancel the common factor of $2$ .

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Step 5.2.4.1

Cancel the common factor.

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

Step 5.2.4.2

Rewrite the expression.

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

Step 5.2.5

Simplify.

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