Finite Math Examples

$\frac{9 - \sqrt{2}}{2 + 3 \sqrt{2}}$

Step 1

Multiply $\frac{9 - \sqrt{2}}{2 + 3 \sqrt{2}}$ by $\frac{2 - 3 \sqrt{2}}{2 - 3 \sqrt{2}}$ .

$\frac{9 - \sqrt{2}}{2 + 3 \sqrt{2}} \cdot \frac{2 - 3 \sqrt{2}}{2 - 3 \sqrt{2}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{9 - \sqrt{2}}{2 + 3 \sqrt{2}}$ by $\frac{2 - 3 \sqrt{2}}{2 - 3 \sqrt{2}}$ .

$\frac{(9 - \sqrt{2}) (2 - 3 \sqrt{2})}{(2 + 3 \sqrt{2}) (2 - 3 \sqrt{2})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(9 - \sqrt{2}) (2 - 3 \sqrt{2})}{4 - 6 \sqrt{2} + 6 \sqrt{2} - 9 {\sqrt{2}}^{2}}$

Step 2.3

Simplify.

$\frac{(9 - \sqrt{2}) (2 - 3 \sqrt{2})}{- 14}$

Step 3

Expand $(9 - \sqrt{2}) (2 - 3 \sqrt{2})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{9 (2 - 3 \sqrt{2}) - \sqrt{2} (2 - 3 \sqrt{2})}{- 14}$

Step 3.2

Apply the distributive property.

$\frac{9 \cdot 2 + 9 (- 3 \sqrt{2}) - \sqrt{2} (2 - 3 \sqrt{2})}{- 14}$

Step 3.3

Apply the distributive property.

$\frac{9 \cdot 2 + 9 (- 3 \sqrt{2}) - \sqrt{2} \cdot 2 - \sqrt{2} (- 3 \sqrt{2})}{- 14}$

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 $9$ by $2$ .

$\frac{18 + 9 (- 3 \sqrt{2}) - \sqrt{2} \cdot 2 - \sqrt{2} (- 3 \sqrt{2})}{- 14}$

Step 4.1.2

Multiply $- 3$ by $9$ .

$\frac{18 - 27 \sqrt{2} - \sqrt{2} \cdot 2 - \sqrt{2} (- 3 \sqrt{2})}{- 14}$

Step 4.1.3

Multiply $2$ by $- 1$ .

$\frac{18 - 27 \sqrt{2} - 2 \sqrt{2} - \sqrt{2} (- 3 \sqrt{2})}{- 14}$

Step 4.1.4

Multiply $- \sqrt{2} (- 3 \sqrt{2})$ .

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Step 4.1.4.1

Multiply $- 3$ by $- 1$ .

$\frac{18 - 27 \sqrt{2} - 2 \sqrt{2} + 3 \sqrt{2} \sqrt{2}}{- 14}$

Step 4.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{18 - 27 \sqrt{2} - 2 \sqrt{2} + 3 ({\sqrt{2}}^{1} \sqrt{2})}{- 14}$

Step 4.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{18 - 27 \sqrt{2} - 2 \sqrt{2} + 3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{- 14}$

Step 4.1.4.4

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

$\frac{18 - 27 \sqrt{2} - 2 \sqrt{2} + 3 {\sqrt{2}}^{1 + 1}}{- 14}$

Step 4.1.4.5

Add $1$ and $1$ .

$\frac{18 - 27 \sqrt{2} - 2 \sqrt{2} + 3 {\sqrt{2}}^{2}}{- 14}$

Step 4.1.5

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

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Step 4.1.5.1

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

$\frac{18 - 27 \sqrt{2} - 2 \sqrt{2} + 3 {(2^{\frac{1}{2}})}^{2}}{- 14}$

Step 4.1.5.2

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

$\frac{18 - 27 \sqrt{2} - 2 \sqrt{2} + 3 \cdot 2^{\frac{1}{2} \cdot 2}}{- 14}$

Step 4.1.5.3

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

$\frac{18 - 27 \sqrt{2} - 2 \sqrt{2} + 3 \cdot 2^{\frac{2}{2}}}{- 14}$

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.

$\frac{18 - 27 \sqrt{2} - 2 \sqrt{2} + 3 \cdot 2^{\frac{2}{2}}}{- 14}$

Step 4.1.5.4.2

Rewrite the expression.

$\frac{18 - 27 \sqrt{2} - 2 \sqrt{2} + 3 \cdot 2^{1}}{- 14}$

Step 4.1.5.5

Evaluate the exponent.

$\frac{18 - 27 \sqrt{2} - 2 \sqrt{2} + 3 \cdot 2}{- 14}$

Step 4.1.6

Multiply $3$ by $2$ .

$\frac{18 - 27 \sqrt{2} - 2 \sqrt{2} + 6}{- 14}$

Step 4.2

Add $18$ and $6$ .

$\frac{24 - 27 \sqrt{2} - 2 \sqrt{2}}{- 14}$

Step 4.3

Subtract $2 \sqrt{2}$ from $- 27 \sqrt{2}$ .

$\frac{24 - 29 \sqrt{2}}{- 14}$

Step 5

Move the negative in front of the fraction.

$- \frac{24 - 29 \sqrt{2}}{14}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{24 - 29 \sqrt{2}}{14}$

Decimal Form:

$1.21515666 \dots$