Finite Math Examples

$\frac{1 - \sqrt{6}}{5 + 2 \sqrt{6}}$

Step 1

Multiply $\frac{1 - \sqrt{6}}{5 + 2 \sqrt{6}}$ by $\frac{5 - 2 \sqrt{6}}{5 - 2 \sqrt{6}}$ .

$\frac{1 - \sqrt{6}}{5 + 2 \sqrt{6}} \cdot \frac{5 - 2 \sqrt{6}}{5 - 2 \sqrt{6}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{1 - \sqrt{6}}{5 + 2 \sqrt{6}}$ by $\frac{5 - 2 \sqrt{6}}{5 - 2 \sqrt{6}}$ .

$\frac{(1 - \sqrt{6}) (5 - 2 \sqrt{6})}{(5 + 2 \sqrt{6}) (5 - 2 \sqrt{6})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(1 - \sqrt{6}) (5 - 2 \sqrt{6})}{25 - 10 \sqrt{6} + 10 \sqrt{6} - 4 {\sqrt{6}}^{2}}$

Step 2.3

Simplify.

$\frac{(1 - \sqrt{6}) (5 - 2 \sqrt{6})}{1}$

Step 2.4

Divide $(1 - \sqrt{6}) (5 - 2 \sqrt{6})$ by $1$ .

$(1 - \sqrt{6}) (5 - 2 \sqrt{6})$

Step 3

Expand $(1 - \sqrt{6}) (5 - 2 \sqrt{6})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$1 (5 - 2 \sqrt{6}) - \sqrt{6} (5 - 2 \sqrt{6})$

Step 3.2

Apply the distributive property.

$1 \cdot 5 + 1 (- 2 \sqrt{6}) - \sqrt{6} (5 - 2 \sqrt{6})$

Step 3.3

Apply the distributive property.

$1 \cdot 5 + 1 (- 2 \sqrt{6}) - \sqrt{6} \cdot 5 - \sqrt{6} (- 2 \sqrt{6})$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $5$ by $1$ .

$5 + 1 (- 2 \sqrt{6}) - \sqrt{6} \cdot 5 - \sqrt{6} (- 2 \sqrt{6})$

Step 4.1.2

Multiply $- 2 \sqrt{6}$ by $1$ .

$5 - 2 \sqrt{6} - \sqrt{6} \cdot 5 - \sqrt{6} (- 2 \sqrt{6})$

Step 4.1.3

Multiply $5$ by $- 1$ .

$5 - 2 \sqrt{6} - 5 \sqrt{6} - \sqrt{6} (- 2 \sqrt{6})$

Step 4.1.4

Multiply $- \sqrt{6} (- 2 \sqrt{6})$ .

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Step 4.1.4.1

Multiply $- 2$ by $- 1$ .

$5 - 2 \sqrt{6} - 5 \sqrt{6} + 2 \sqrt{6} \sqrt{6}$

Step 4.1.4.2

Raise $\sqrt{6}$ to the power of $1$ .

$5 - 2 \sqrt{6} - 5 \sqrt{6} + 2 ({\sqrt{6}}^{1} \sqrt{6})$

Step 4.1.4.3

Raise $\sqrt{6}$ to the power of $1$ .

$5 - 2 \sqrt{6} - 5 \sqrt{6} + 2 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})$

Step 4.1.4.4

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

$5 - 2 \sqrt{6} - 5 \sqrt{6} + 2 {\sqrt{6}}^{1 + 1}$

Step 4.1.4.5

Add $1$ and $1$ .

$5 - 2 \sqrt{6} - 5 \sqrt{6} + 2 {\sqrt{6}}^{2}$

Step 4.1.5

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$5 - 2 \sqrt{6} - 5 \sqrt{6} + 2 {(6^{\frac{1}{2}})}^{2}$

Step 4.1.5.2

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

$5 - 2 \sqrt{6} - 5 \sqrt{6} + 2 \cdot 6^{\frac{1}{2} \cdot 2}$

Step 4.1.5.3

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

$5 - 2 \sqrt{6} - 5 \sqrt{6} + 2 \cdot 6^{\frac{2}{2}}$

Step 4.1.5.4

Cancel the common factor of $2$ .

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Step 4.1.5.4.1

Cancel the common factor.

$5 - 2 \sqrt{6} - 5 \sqrt{6} + 2 \cdot 6^{\frac{2}{2}}$

Step 4.1.5.4.2

Rewrite the expression.

$5 - 2 \sqrt{6} - 5 \sqrt{6} + 2 \cdot 6^{1}$

Step 4.1.5.5

Evaluate the exponent.

$5 - 2 \sqrt{6} - 5 \sqrt{6} + 2 \cdot 6$

Step 4.1.6

Multiply $2$ by $6$ .

$5 - 2 \sqrt{6} - 5 \sqrt{6} + 12$

Step 4.2

Add $5$ and $12$ .

$17 - 2 \sqrt{6} - 5 \sqrt{6}$

Step 4.3

Subtract $5 \sqrt{6}$ from $- 2 \sqrt{6}$ .

$17 - 7 \sqrt{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$17 - 7 \sqrt{6}$

Decimal Form:

$- 0.14642819 \dots$