Basic Math Examples

$\frac{1 + \sqrt{a}}{1 - \sqrt{a}}$

Step 1

Multiply $\frac{1 + \sqrt{a}}{1 - \sqrt{a}}$ by $\frac{1 + \sqrt{a}}{1 + \sqrt{a}}$ .

$\frac{1 + \sqrt{a}}{1 - \sqrt{a}} \cdot \frac{1 + \sqrt{a}}{1 + \sqrt{a}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{1 + \sqrt{a}}{1 - \sqrt{a}}$ by $\frac{1 + \sqrt{a}}{1 + \sqrt{a}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

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

Step 2.3.4.2

Rewrite the expression.

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

Step 2.3.5

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $1$ .

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

Step 4

Rewrite ${(1 + \sqrt{a})}^{2}$ as $(1 + \sqrt{a}) (1 + \sqrt{a})$ .

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

Step 5

Expand $(1 + \sqrt{a}) (1 + \sqrt{a})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

$\frac{1 \cdot 1 + 1 \sqrt{a} + \sqrt{a} \cdot 1 + \sqrt{a} \sqrt{a}}{1 - a}$

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 $1$ by $1$ .

$\frac{1 + 1 \sqrt{a} + \sqrt{a} \cdot 1 + \sqrt{a} \sqrt{a}}{1 - a}$

Step 6.1.2

Multiply $\sqrt{a}$ by $1$ .

$\frac{1 + \sqrt{a} + \sqrt{a} \cdot 1 + \sqrt{a} \sqrt{a}}{1 - a}$

Step 6.1.3

Multiply $\sqrt{a}$ by $1$ .

$\frac{1 + \sqrt{a} + \sqrt{a} + \sqrt{a} \sqrt{a}}{1 - a}$

Step 6.1.4

Multiply $\sqrt{a} \sqrt{a}$ .

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Step 6.1.4.1

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

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

Step 6.1.4.2

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

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

Step 6.1.4.3

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

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

Step 6.1.4.4

Add $1$ and $1$ .

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

Step 6.1.5

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

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Step 6.1.5.1

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

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

Step 6.1.5.2

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

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

Step 6.1.5.3

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

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

Step 6.1.5.4

Cancel the common factor of $2$ .

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Step 6.1.5.4.1

Cancel the common factor.

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

Step 6.1.5.4.2

Rewrite the expression.

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

Step 6.1.5.5

Simplify.

$\frac{1 + \sqrt{a} + \sqrt{a} + a}{1 - a}$

Step 6.2

Add $\sqrt{a}$ and $\sqrt{a}$ .

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