Trigonometry Examples

$\frac{- 15 - 4 \sqrt{11}}{20 - 3 \sqrt{11}}$

Step 1

Multiply $\frac{- 15 - 4 \sqrt{11}}{20 - 3 \sqrt{11}}$ by $\frac{20 + 3 \sqrt{11}}{20 + 3 \sqrt{11}}$ .

$\frac{- 15 - 4 \sqrt{11}}{20 - 3 \sqrt{11}} \cdot \frac{20 + 3 \sqrt{11}}{20 + 3 \sqrt{11}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{- 15 - 4 \sqrt{11}}{20 - 3 \sqrt{11}}$ by $\frac{20 + 3 \sqrt{11}}{20 + 3 \sqrt{11}}$ .

$\frac{(- 15 - 4 \sqrt{11}) (20 + 3 \sqrt{11})}{(20 - 3 \sqrt{11}) (20 + 3 \sqrt{11})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(- 15 - 4 \sqrt{11}) (20 + 3 \sqrt{11})}{400 + 60 \sqrt{11} - 60 \sqrt{11} - 9 {\sqrt{11}}^{2}}$

Step 2.3

Simplify.

$\frac{(- 15 - 4 \sqrt{11}) (20 + 3 \sqrt{11})}{301}$

Step 3

Expand $(- 15 - 4 \sqrt{11}) (20 + 3 \sqrt{11})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{- 15 (20 + 3 \sqrt{11}) - 4 \sqrt{11} (20 + 3 \sqrt{11})}{301}$

Step 3.2

Apply the distributive property.

$\frac{- 15 \cdot 20 - 15 (3 \sqrt{11}) - 4 \sqrt{11} (20 + 3 \sqrt{11})}{301}$

Step 3.3

Apply the distributive property.

$\frac{- 15 \cdot 20 - 15 (3 \sqrt{11}) - 4 \sqrt{11} \cdot 20 - 4 \sqrt{11} (3 \sqrt{11})}{301}$

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 $- 15$ by $20$ .

$\frac{- 300 - 15 (3 \sqrt{11}) - 4 \sqrt{11} \cdot 20 - 4 \sqrt{11} (3 \sqrt{11})}{301}$

Step 4.1.2

Multiply $3$ by $- 15$ .

$\frac{- 300 - 45 \sqrt{11} - 4 \sqrt{11} \cdot 20 - 4 \sqrt{11} (3 \sqrt{11})}{301}$

Step 4.1.3

Multiply $20$ by $- 4$ .

$\frac{- 300 - 45 \sqrt{11} - 80 \sqrt{11} - 4 \sqrt{11} (3 \sqrt{11})}{301}$

Step 4.1.4

Multiply $- 4 \sqrt{11} (3 \sqrt{11})$ .

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Step 4.1.4.1

Multiply $3$ by $- 4$ .

$\frac{- 300 - 45 \sqrt{11} - 80 \sqrt{11} - 12 \sqrt{11} \sqrt{11}}{301}$

Step 4.1.4.2

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{- 300 - 45 \sqrt{11} - 80 \sqrt{11} - 12 ({\sqrt{11}}^{1} \sqrt{11})}{301}$

Step 4.1.4.3

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{- 300 - 45 \sqrt{11} - 80 \sqrt{11} - 12 ({\sqrt{11}}^{1} {\sqrt{11}}^{1})}{301}$

Step 4.1.4.4

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

$\frac{- 300 - 45 \sqrt{11} - 80 \sqrt{11} - 12 {\sqrt{11}}^{1 + 1}}{301}$

Step 4.1.4.5

Add $1$ and $1$ .

$\frac{- 300 - 45 \sqrt{11} - 80 \sqrt{11} - 12 {\sqrt{11}}^{2}}{301}$

Step 4.1.5

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 4.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{- 300 - 45 \sqrt{11} - 80 \sqrt{11} - 12 {(11^{\frac{1}{2}})}^{2}}{301}$

Step 4.1.5.2

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

$\frac{- 300 - 45 \sqrt{11} - 80 \sqrt{11} - 12 \cdot 11^{\frac{1}{2} \cdot 2}}{301}$

Step 4.1.5.3

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

$\frac{- 300 - 45 \sqrt{11} - 80 \sqrt{11} - 12 \cdot 11^{\frac{2}{2}}}{301}$

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.

$\frac{- 300 - 45 \sqrt{11} - 80 \sqrt{11} - 12 \cdot 11^{\frac{2}{2}}}{301}$

Step 4.1.5.4.2

Rewrite the expression.

$\frac{- 300 - 45 \sqrt{11} - 80 \sqrt{11} - 12 \cdot 11^{1}}{301}$

Step 4.1.5.5

Evaluate the exponent.

$\frac{- 300 - 45 \sqrt{11} - 80 \sqrt{11} - 12 \cdot 11}{301}$

Step 4.1.6

Multiply $- 12$ by $11$ .

$\frac{- 300 - 45 \sqrt{11} - 80 \sqrt{11} - 132}{301}$

Step 4.2

Subtract $132$ from $- 300$ .

$\frac{- 432 - 45 \sqrt{11} - 80 \sqrt{11}}{301}$

Step 4.3

Subtract $80 \sqrt{11}$ from $- 45 \sqrt{11}$ .

$\frac{- 432 - 125 \sqrt{11}}{301}$

Step 5

Rewrite $- 432$ as $- 1 (432)$ .

$\frac{- 1 (432) - 125 \sqrt{11}}{301}$

Step 6

Factor $- 1$ out of $- 125 \sqrt{11}$ .

$\frac{- 1 (432) - (125 \sqrt{11})}{301}$

Step 7

Factor $- 1$ out of $- 1 (432) - (125 \sqrt{11})$ .

$\frac{- 1 (432 + 125 \sqrt{11})}{301}$

Step 8

Move the negative in front of the fraction.

$- \frac{432 + 125 \sqrt{11}}{301}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{432 + 125 \sqrt{11}}{301}$

Decimal Form:

$- 2.81255182 \dots$