Trigonometry Examples

$\frac{- 9 - 4 \sqrt{55}}{12 - 3 \sqrt{55}}$

Step 1

Multiply $\frac{- 9 - 4 \sqrt{55}}{12 - 3 \sqrt{55}}$ by $\frac{12 + 3 \sqrt{55}}{12 + 3 \sqrt{55}}$ .

$\frac{- 9 - 4 \sqrt{55}}{12 - 3 \sqrt{55}} \cdot \frac{12 + 3 \sqrt{55}}{12 + 3 \sqrt{55}}$

Step 2

Simplify terms.

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Step 2.1

Multiply $\frac{- 9 - 4 \sqrt{55}}{12 - 3 \sqrt{55}}$ by $\frac{12 + 3 \sqrt{55}}{12 + 3 \sqrt{55}}$ .

$\frac{(- 9 - 4 \sqrt{55}) (12 + 3 \sqrt{55})}{(12 - 3 \sqrt{55}) (12 + 3 \sqrt{55})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(- 9 - 4 \sqrt{55}) (12 + 3 \sqrt{55})}{144 + 36 \sqrt{55} - 36 \sqrt{55} - 9 {\sqrt{55}}^{2}}$

Step 2.3

Simplify.

$\frac{(- 9 - 4 \sqrt{55}) (12 + 3 \sqrt{55})}{- 351}$

Step 2.4

Cancel the common factor of $12 + 3 \sqrt{55}$ and $- 351$ .

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Step 2.4.1

Factor $3$ out of $(- 9 - 4 \sqrt{55}) (12 + 3 \sqrt{55})$ .

$\frac{3 ((- 9 - 4 \sqrt{55}) (4 + \sqrt{55}))}{- 351}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $3$ out of $- 351$ .

$\frac{3 ((- 9 - 4 \sqrt{55}) (4 + \sqrt{55}))}{3 (- 117)}$

Step 2.4.2.2

Cancel the common factor.

$\frac{3 ((- 9 - 4 \sqrt{55}) (4 + \sqrt{55}))}{3 \cdot - 117}$

Step 2.4.2.3

Rewrite the expression.

$\frac{(- 9 - 4 \sqrt{55}) (4 + \sqrt{55})}{- 117}$

Step 3

Expand $(- 9 - 4 \sqrt{55}) (4 + \sqrt{55})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{- 9 (4 + \sqrt{55}) - 4 \sqrt{55} (4 + \sqrt{55})}{- 117}$

Step 3.2

Apply the distributive property.

$\frac{- 9 \cdot 4 - 9 \sqrt{55} - 4 \sqrt{55} (4 + \sqrt{55})}{- 117}$

Step 3.3

Apply the distributive property.

$\frac{- 9 \cdot 4 - 9 \sqrt{55} - 4 \sqrt{55} \cdot 4 - 4 \sqrt{55} \sqrt{55}}{- 117}$

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 $4$ .

$\frac{- 36 - 9 \sqrt{55} - 4 \sqrt{55} \cdot 4 - 4 \sqrt{55} \sqrt{55}}{- 117}$

Step 4.1.2

Multiply $4$ by $- 4$ .

$\frac{- 36 - 9 \sqrt{55} - 16 \sqrt{55} - 4 \sqrt{55} \sqrt{55}}{- 117}$

Step 4.1.3

Multiply $- 4 \sqrt{55} \sqrt{55}$ .

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Step 4.1.3.1

Raise $\sqrt{55}$ to the power of $1$ .

$\frac{- 36 - 9 \sqrt{55} - 16 \sqrt{55} - 4 ({\sqrt{55}}^{1} \sqrt{55})}{- 117}$

Step 4.1.3.2

Raise $\sqrt{55}$ to the power of $1$ .

$\frac{- 36 - 9 \sqrt{55} - 16 \sqrt{55} - 4 ({\sqrt{55}}^{1} {\sqrt{55}}^{1})}{- 117}$

Step 4.1.3.3

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

$\frac{- 36 - 9 \sqrt{55} - 16 \sqrt{55} - 4 {\sqrt{55}}^{1 + 1}}{- 117}$

Step 4.1.3.4

Add $1$ and $1$ .

$\frac{- 36 - 9 \sqrt{55} - 16 \sqrt{55} - 4 {\sqrt{55}}^{2}}{- 117}$

Step 4.1.4

Rewrite ${\sqrt{55}}^{2}$ as $55$ .

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Step 4.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{55}$ as $55^{\frac{1}{2}}$ .

$\frac{- 36 - 9 \sqrt{55} - 16 \sqrt{55} - 4 {(55^{\frac{1}{2}})}^{2}}{- 117}$

Step 4.1.4.2

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

$\frac{- 36 - 9 \sqrt{55} - 16 \sqrt{55} - 4 \cdot 55^{\frac{1}{2} \cdot 2}}{- 117}$

Step 4.1.4.3

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

$\frac{- 36 - 9 \sqrt{55} - 16 \sqrt{55} - 4 \cdot 55^{\frac{2}{2}}}{- 117}$

Step 4.1.4.4

Cancel the common factor of $2$ .

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Step 4.1.4.4.1

Cancel the common factor.

$\frac{- 36 - 9 \sqrt{55} - 16 \sqrt{55} - 4 \cdot 55^{\frac{2}{2}}}{- 117}$

Step 4.1.4.4.2

Rewrite the expression.

$\frac{- 36 - 9 \sqrt{55} - 16 \sqrt{55} - 4 \cdot 55^{1}}{- 117}$

Step 4.1.4.5

Evaluate the exponent.

$\frac{- 36 - 9 \sqrt{55} - 16 \sqrt{55} - 4 \cdot 55}{- 117}$

Step 4.1.5

Multiply $- 4$ by $55$ .

$\frac{- 36 - 9 \sqrt{55} - 16 \sqrt{55} - 220}{- 117}$

Step 4.2

Subtract $220$ from $- 36$ .

$\frac{- 256 - 9 \sqrt{55} - 16 \sqrt{55}}{- 117}$

Step 4.3

Subtract $16 \sqrt{55}$ from $- 9 \sqrt{55}$ .

$\frac{- 256 - 25 \sqrt{55}}{- 117}$

Step 5

Move the negative in front of the fraction.

$- \frac{- 256 - 25 \sqrt{55}}{117}$

Step 6

Rewrite $- 256$ as $- 1 (256)$ .

$- \frac{- 1 (256) - 25 \sqrt{55}}{117}$

Step 7

Factor $- 1$ out of $- 25 \sqrt{55}$ .

$- \frac{- 1 (256) - (25 \sqrt{55})}{117}$

Step 8

Factor $- 1$ out of $- 1 (256) - (25 \sqrt{55})$ .

$- \frac{- 1 (256 + 25 \sqrt{55})}{117}$

Step 9

Simplify the expression.

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Step 9.1

Move the negative in front of the fraction.

$- - \frac{256 + 25 \sqrt{55}}{117}$

Step 9.2

Multiply $- 1$ by $- 1$ .

$1 \frac{256 + 25 \sqrt{55}}{117}$

Step 9.3

Multiply $\frac{256 + 25 \sqrt{55}}{117}$ by $1$ .

$\frac{256 + 25 \sqrt{55}}{117}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{256 + 25 \sqrt{55}}{117}$

Decimal Form:

$3.77269198 \dots$