Linear Algebra Examples

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

Step 1

Multiply $\frac{a - b}{\sqrt{a} - \sqrt{b}}$ by $\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} + \sqrt{b}}$ .

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

Step 2

Multiply $\frac{a - b}{\sqrt{a} - \sqrt{b}}$ by $\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} + \sqrt{b}}$ .

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Cancel the common factor.

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

Step 4.1.4.2

Rewrite the expression.

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

Step 4.1.5

Simplify.

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

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

Step 4.2.4.2

Rewrite the expression.

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

Step 4.2.5

Simplify.

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

Step 5

Cancel the common factor of $a - b$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Divide $\sqrt{a} + \sqrt{b}$ by $1$ .

$\sqrt{a} + \sqrt{b}$