Finite Math Examples

$\frac{5 \sqrt{3} - 3 \sqrt{2}}{3 \sqrt{2} - 2 \sqrt{3}}$

Step 1

Multiply $\frac{5 \sqrt{3} - 3 \sqrt{2}}{3 \sqrt{2} - 2 \sqrt{3}}$ by $\frac{3 \sqrt{2} + 2 \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}}$ .

$\frac{5 \sqrt{3} - 3 \sqrt{2}}{3 \sqrt{2} - 2 \sqrt{3}} \cdot \frac{3 \sqrt{2} + 2 \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{5 \sqrt{3} - 3 \sqrt{2}}{3 \sqrt{2} - 2 \sqrt{3}}$ by $\frac{3 \sqrt{2} + 2 \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}}$ .

$\frac{(5 \sqrt{3} - 3 \sqrt{2}) (3 \sqrt{2} + 2 \sqrt{3})}{(3 \sqrt{2} - 2 \sqrt{3}) (3 \sqrt{2} + 2 \sqrt{3})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(5 \sqrt{3} - 3 \sqrt{2}) (3 \sqrt{2} + 2 \sqrt{3})}{9 {\sqrt{2}}^{2} + 6 \sqrt{6} - 6 \sqrt{6} - 4 {\sqrt{3}}^{2}}$

Step 2.3

Simplify.

$\frac{(5 \sqrt{3} - 3 \sqrt{2}) (3 \sqrt{2} + 2 \sqrt{3})}{6}$

Step 3

Expand $(5 \sqrt{3} - 3 \sqrt{2}) (3 \sqrt{2} + 2 \sqrt{3})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{5 \sqrt{3} (3 \sqrt{2} + 2 \sqrt{3}) - 3 \sqrt{2} (3 \sqrt{2} + 2 \sqrt{3})}{6}$

Step 3.2

Apply the distributive property.

$\frac{5 \sqrt{3} (3 \sqrt{2}) + 5 \sqrt{3} (2 \sqrt{3}) - 3 \sqrt{2} (3 \sqrt{2} + 2 \sqrt{3})}{6}$

Step 3.3

Apply the distributive property.

$\frac{5 \sqrt{3} (3 \sqrt{2}) + 5 \sqrt{3} (2 \sqrt{3}) - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

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 $5 \sqrt{3} (3 \sqrt{2})$ .

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Step 4.1.1.1

Multiply $3$ by $5$ .

$\frac{15 \sqrt{3} \sqrt{2} + 5 \sqrt{3} (2 \sqrt{3}) - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.1.2

Combine using the product rule for radicals.

$\frac{15 \sqrt{2 \cdot 3} + 5 \sqrt{3} (2 \sqrt{3}) - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.1.3

Multiply $2$ by $3$ .

$\frac{15 \sqrt{6} + 5 \sqrt{3} (2 \sqrt{3}) - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.2

Multiply $5 \sqrt{3} (2 \sqrt{3})$ .

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Step 4.1.2.1

Multiply $2$ by $5$ .

$\frac{15 \sqrt{6} + 10 \sqrt{3} \sqrt{3} - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.2.2

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

$\frac{15 \sqrt{6} + 10 ({\sqrt{3}}^{1} \sqrt{3}) - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.2.3

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

$\frac{15 \sqrt{6} + 10 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.2.4

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

$\frac{15 \sqrt{6} + 10 {\sqrt{3}}^{1 + 1} - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.2.5

Add $1$ and $1$ .

$\frac{15 \sqrt{6} + 10 {\sqrt{3}}^{2} - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.3

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

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Step 4.1.3.1

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

$\frac{15 \sqrt{6} + 10 {(3^{\frac{1}{2}})}^{2} - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.3.2

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

$\frac{15 \sqrt{6} + 10 \cdot 3^{\frac{1}{2} \cdot 2} - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.3.3

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

$\frac{15 \sqrt{6} + 10 \cdot 3^{\frac{2}{2}} - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.3.4

Cancel the common factor of $2$ .

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Step 4.1.3.4.1

Cancel the common factor.

$\frac{15 \sqrt{6} + 10 \cdot 3^{\frac{2}{2}} - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.3.4.2

Rewrite the expression.

$\frac{15 \sqrt{6} + 10 \cdot 3^{1} - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.3.5

Evaluate the exponent.

$\frac{15 \sqrt{6} + 10 \cdot 3 - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.4

Multiply $10$ by $3$ .

$\frac{15 \sqrt{6} + 30 - 3 \sqrt{2} (3 \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.5

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

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Step 4.1.5.1

Multiply $3$ by $- 3$ .

$\frac{15 \sqrt{6} + 30 - 9 \sqrt{2} \sqrt{2} - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.5.2

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

$\frac{15 \sqrt{6} + 30 - 9 ({\sqrt{2}}^{1} \sqrt{2}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.5.3

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

$\frac{15 \sqrt{6} + 30 - 9 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.5.4

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

$\frac{15 \sqrt{6} + 30 - 9 {\sqrt{2}}^{1 + 1} - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.5.5

Add $1$ and $1$ .

$\frac{15 \sqrt{6} + 30 - 9 {\sqrt{2}}^{2} - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.6

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

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Step 4.1.6.1

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

$\frac{15 \sqrt{6} + 30 - 9 {(2^{\frac{1}{2}})}^{2} - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.6.2

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

$\frac{15 \sqrt{6} + 30 - 9 \cdot 2^{\frac{1}{2} \cdot 2} - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.6.3

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

$\frac{15 \sqrt{6} + 30 - 9 \cdot 2^{\frac{2}{2}} - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.6.4

Cancel the common factor of $2$ .

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Step 4.1.6.4.1

Cancel the common factor.

$\frac{15 \sqrt{6} + 30 - 9 \cdot 2^{\frac{2}{2}} - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.6.4.2

Rewrite the expression.

$\frac{15 \sqrt{6} + 30 - 9 \cdot 2^{1} - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.6.5

Evaluate the exponent.

$\frac{15 \sqrt{6} + 30 - 9 \cdot 2 - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.7

Multiply $- 9$ by $2$ .

$\frac{15 \sqrt{6} + 30 - 18 - 3 \sqrt{2} (2 \sqrt{3})}{6}$

Step 4.1.8

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

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Step 4.1.8.1

Multiply $2$ by $- 3$ .

$\frac{15 \sqrt{6} + 30 - 18 - 6 \sqrt{2} \sqrt{3}}{6}$

Step 4.1.8.2

Combine using the product rule for radicals.

$\frac{15 \sqrt{6} + 30 - 18 - 6 \sqrt{3 \cdot 2}}{6}$

Step 4.1.8.3

Multiply $3$ by $2$ .

$\frac{15 \sqrt{6} + 30 - 18 - 6 \sqrt{6}}{6}$

Step 4.2

Subtract $6 \sqrt{6}$ from $15 \sqrt{6}$ .

$\frac{30 - 18 + 9 \sqrt{6}}{6}$

Step 4.3

Subtract $18$ from $30$ .

$\frac{12 + 9 \sqrt{6}}{6}$

Step 5

Cancel the common factor of $12 + 9 \sqrt{6}$ and $6$ .

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Step 5.1

Factor $3$ out of $12$ .

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

Step 5.2

Factor $3$ out of $9 \sqrt{6}$ .

$\frac{3 (4) + 3 (3 \sqrt{6})}{6}$

Step 5.3

Factor $3$ out of $3 (4) + 3 (3 \sqrt{6})$ .

$\frac{3 (4 + 3 \sqrt{6})}{6}$

Step 5.4

Cancel the common factors.

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Step 5.4.1

Factor $3$ out of $6$ .

$\frac{3 (4 + 3 \sqrt{6})}{3 \cdot 2}$

Step 5.4.2

Cancel the common factor.

$\frac{3 (4 + 3 \sqrt{6})}{3 \cdot 2}$

Step 5.4.3

Rewrite the expression.

$\frac{4 + 3 \sqrt{6}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{4 + 3 \sqrt{6}}{2}$

Decimal Form:

$5.67423461 \dots$