Finite Math Examples

$\frac{\sqrt{3} - 2 \sqrt{2}}{3 \sqrt{3} + 7 \sqrt{2}}$

Step 1

Multiply $\frac{\sqrt{3} - 2 \sqrt{2}}{3 \sqrt{3} + 7 \sqrt{2}}$ by $\frac{3 \sqrt{3} - 7 \sqrt{2}}{3 \sqrt{3} - 7 \sqrt{2}}$ .

$\frac{\sqrt{3} - 2 \sqrt{2}}{3 \sqrt{3} + 7 \sqrt{2}} \cdot \frac{3 \sqrt{3} - 7 \sqrt{2}}{3 \sqrt{3} - 7 \sqrt{2}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{3} - 2 \sqrt{2}}{3 \sqrt{3} + 7 \sqrt{2}}$ by $\frac{3 \sqrt{3} - 7 \sqrt{2}}{3 \sqrt{3} - 7 \sqrt{2}}$ .

$\frac{(\sqrt{3} - 2 \sqrt{2}) (3 \sqrt{3} - 7 \sqrt{2})}{(3 \sqrt{3} + 7 \sqrt{2}) (3 \sqrt{3} - 7 \sqrt{2})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{3} - 2 \sqrt{2}) (3 \sqrt{3} - 7 \sqrt{2})}{9 {\sqrt{3}}^{2} - 21 \sqrt{6} + 21 \sqrt{6} - 49 {\sqrt{2}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{3} - 2 \sqrt{2}) (3 \sqrt{3} - 7 \sqrt{2})}{- 71}$

Step 3

Expand $(\sqrt{3} - 2 \sqrt{2}) (3 \sqrt{3} - 7 \sqrt{2})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{3} (3 \sqrt{3} - 7 \sqrt{2}) - 2 \sqrt{2} (3 \sqrt{3} - 7 \sqrt{2})}{- 71}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{3} (3 \sqrt{3}) + \sqrt{3} (- 7 \sqrt{2}) - 2 \sqrt{2} (3 \sqrt{3} - 7 \sqrt{2})}{- 71}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{3} (3 \sqrt{3}) + \sqrt{3} (- 7 \sqrt{2}) - 2 \sqrt{2} (3 \sqrt{3}) - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

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 $\sqrt{3} (3 \sqrt{3})$ .

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Step 4.1.1.1

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{3 ({\sqrt{3}}^{1} \sqrt{3}) + \sqrt{3} (- 7 \sqrt{2}) - 2 \sqrt{2} (3 \sqrt{3}) - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.1.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{3 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) + \sqrt{3} (- 7 \sqrt{2}) - 2 \sqrt{2} (3 \sqrt{3}) - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.1.3

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

$\frac{3 {\sqrt{3}}^{1 + 1} + \sqrt{3} (- 7 \sqrt{2}) - 2 \sqrt{2} (3 \sqrt{3}) - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.1.4

Add $1$ and $1$ .

$\frac{3 {\sqrt{3}}^{2} + \sqrt{3} (- 7 \sqrt{2}) - 2 \sqrt{2} (3 \sqrt{3}) - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{3 {(3^{\frac{1}{2}})}^{2} + \sqrt{3} (- 7 \sqrt{2}) - 2 \sqrt{2} (3 \sqrt{3}) - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.2.2

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

$\frac{3 \cdot 3^{\frac{1}{2} \cdot 2} + \sqrt{3} (- 7 \sqrt{2}) - 2 \sqrt{2} (3 \sqrt{3}) - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.2.3

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

$\frac{3 \cdot 3^{\frac{2}{2}} + \sqrt{3} (- 7 \sqrt{2}) - 2 \sqrt{2} (3 \sqrt{3}) - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.4.1

Cancel the common factor.

$\frac{3 \cdot 3^{\frac{2}{2}} + \sqrt{3} (- 7 \sqrt{2}) - 2 \sqrt{2} (3 \sqrt{3}) - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.2.4.2

Rewrite the expression.

$\frac{3 \cdot 3^{1} + \sqrt{3} (- 7 \sqrt{2}) - 2 \sqrt{2} (3 \sqrt{3}) - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.2.5

Evaluate the exponent.

$\frac{3 \cdot 3 + \sqrt{3} (- 7 \sqrt{2}) - 2 \sqrt{2} (3 \sqrt{3}) - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.3

Multiply $3$ by $3$ .

$\frac{9 + \sqrt{3} (- 7 \sqrt{2}) - 2 \sqrt{2} (3 \sqrt{3}) - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.4

Multiply $\sqrt{3} (- 7 \sqrt{2})$ .

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Step 4.1.4.1

Combine using the product rule for radicals.

$\frac{9 - 7 \sqrt{3 \cdot 2} - 2 \sqrt{2} (3 \sqrt{3}) - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.4.2

Multiply $3$ by $2$ .

$\frac{9 - 7 \sqrt{6} - 2 \sqrt{2} (3 \sqrt{3}) - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.5

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

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Step 4.1.5.1

Multiply $3$ by $- 2$ .

$\frac{9 - 7 \sqrt{6} - 6 \sqrt{2} \sqrt{3} - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.5.2

Combine using the product rule for radicals.

$\frac{9 - 7 \sqrt{6} - 6 \sqrt{3 \cdot 2} - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.5.3

Multiply $3$ by $2$ .

$\frac{9 - 7 \sqrt{6} - 6 \sqrt{6} - 2 \sqrt{2} (- 7 \sqrt{2})}{- 71}$

Step 4.1.6

Multiply $- 2 \sqrt{2} (- 7 \sqrt{2})$ .

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Step 4.1.6.1

Multiply $- 7$ by $- 2$ .

$\frac{9 - 7 \sqrt{6} - 6 \sqrt{6} + 14 \sqrt{2} \sqrt{2}}{- 71}$

Step 4.1.6.2

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

$\frac{9 - 7 \sqrt{6} - 6 \sqrt{6} + 14 ({\sqrt{2}}^{1} \sqrt{2})}{- 71}$

Step 4.1.6.3

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

$\frac{9 - 7 \sqrt{6} - 6 \sqrt{6} + 14 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{- 71}$

Step 4.1.6.4

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

$\frac{9 - 7 \sqrt{6} - 6 \sqrt{6} + 14 {\sqrt{2}}^{1 + 1}}{- 71}$

Step 4.1.6.5

Add $1$ and $1$ .

$\frac{9 - 7 \sqrt{6} - 6 \sqrt{6} + 14 {\sqrt{2}}^{2}}{- 71}$

Step 4.1.7

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

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Step 4.1.7.1

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

$\frac{9 - 7 \sqrt{6} - 6 \sqrt{6} + 14 {(2^{\frac{1}{2}})}^{2}}{- 71}$

Step 4.1.7.2

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

$\frac{9 - 7 \sqrt{6} - 6 \sqrt{6} + 14 \cdot 2^{\frac{1}{2} \cdot 2}}{- 71}$

Step 4.1.7.3

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

$\frac{9 - 7 \sqrt{6} - 6 \sqrt{6} + 14 \cdot 2^{\frac{2}{2}}}{- 71}$

Step 4.1.7.4

Cancel the common factor of $2$ .

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Step 4.1.7.4.1

Cancel the common factor.

$\frac{9 - 7 \sqrt{6} - 6 \sqrt{6} + 14 \cdot 2^{\frac{2}{2}}}{- 71}$

Step 4.1.7.4.2

Rewrite the expression.

$\frac{9 - 7 \sqrt{6} - 6 \sqrt{6} + 14 \cdot 2^{1}}{- 71}$

Step 4.1.7.5

Evaluate the exponent.

$\frac{9 - 7 \sqrt{6} - 6 \sqrt{6} + 14 \cdot 2}{- 71}$

Step 4.1.8

Multiply $14$ by $2$ .

$\frac{9 - 7 \sqrt{6} - 6 \sqrt{6} + 28}{- 71}$

Step 4.2

Add $9$ and $28$ .

$\frac{37 - 7 \sqrt{6} - 6 \sqrt{6}}{- 71}$

Step 4.3

Subtract $6 \sqrt{6}$ from $- 7 \sqrt{6}$ .

$\frac{37 - 13 \sqrt{6}}{- 71}$

Step 5

Move the negative in front of the fraction.

$- \frac{37 - 13 \sqrt{6}}{71}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{37 - 13 \sqrt{6}}{71}$

Decimal Form:

$- 0.07262863 \dots$