Trigonometry Examples

$\frac{- 56 + 15 \sqrt{15}}{105 + 8 \sqrt{15}}$

Step 1

Multiply $\frac{- 56 + 15 \sqrt{15}}{105 + 8 \sqrt{15}}$ by $\frac{105 - 8 \sqrt{15}}{105 - 8 \sqrt{15}}$ .

$\frac{- 56 + 15 \sqrt{15}}{105 + 8 \sqrt{15}} \cdot \frac{105 - 8 \sqrt{15}}{105 - 8 \sqrt{15}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{- 56 + 15 \sqrt{15}}{105 + 8 \sqrt{15}}$ by $\frac{105 - 8 \sqrt{15}}{105 - 8 \sqrt{15}}$ .

$\frac{(- 56 + 15 \sqrt{15}) (105 - 8 \sqrt{15})}{(105 + 8 \sqrt{15}) (105 - 8 \sqrt{15})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(- 56 + 15 \sqrt{15}) (105 - 8 \sqrt{15})}{11025 - 840 \sqrt{15} + 840 \sqrt{15} - 64 {\sqrt{15}}^{2}}$

Step 2.3

Simplify.

$\frac{(- 56 + 15 \sqrt{15}) (105 - 8 \sqrt{15})}{10065}$

Step 3

Expand $(- 56 + 15 \sqrt{15}) (105 - 8 \sqrt{15})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{- 56 (105 - 8 \sqrt{15}) + 15 \sqrt{15} (105 - 8 \sqrt{15})}{10065}$

Step 3.2

Apply the distributive property.

$\frac{- 56 \cdot 105 - 56 (- 8 \sqrt{15}) + 15 \sqrt{15} (105 - 8 \sqrt{15})}{10065}$

Step 3.3

Apply the distributive property.

$\frac{- 56 \cdot 105 - 56 (- 8 \sqrt{15}) + 15 \sqrt{15} \cdot 105 + 15 \sqrt{15} (- 8 \sqrt{15})}{10065}$

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 $- 56$ by $105$ .

$\frac{- 5880 - 56 (- 8 \sqrt{15}) + 15 \sqrt{15} \cdot 105 + 15 \sqrt{15} (- 8 \sqrt{15})}{10065}$

Step 4.1.2

Multiply $- 8$ by $- 56$ .

$\frac{- 5880 + 448 \sqrt{15} + 15 \sqrt{15} \cdot 105 + 15 \sqrt{15} (- 8 \sqrt{15})}{10065}$

Step 4.1.3

Multiply $105$ by $15$ .

$\frac{- 5880 + 448 \sqrt{15} + 1575 \sqrt{15} + 15 \sqrt{15} (- 8 \sqrt{15})}{10065}$

Step 4.1.4

Multiply $15 \sqrt{15} (- 8 \sqrt{15})$ .

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Step 4.1.4.1

Multiply $- 8$ by $15$ .

$\frac{- 5880 + 448 \sqrt{15} + 1575 \sqrt{15} - 120 \sqrt{15} \sqrt{15}}{10065}$

Step 4.1.4.2

Raise $\sqrt{15}$ to the power of $1$ .

$\frac{- 5880 + 448 \sqrt{15} + 1575 \sqrt{15} - 120 ({\sqrt{15}}^{1} \sqrt{15})}{10065}$

Step 4.1.4.3

Raise $\sqrt{15}$ to the power of $1$ .

$\frac{- 5880 + 448 \sqrt{15} + 1575 \sqrt{15} - 120 ({\sqrt{15}}^{1} {\sqrt{15}}^{1})}{10065}$

Step 4.1.4.4

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

$\frac{- 5880 + 448 \sqrt{15} + 1575 \sqrt{15} - 120 {\sqrt{15}}^{1 + 1}}{10065}$

Step 4.1.4.5

Add $1$ and $1$ .

$\frac{- 5880 + 448 \sqrt{15} + 1575 \sqrt{15} - 120 {\sqrt{15}}^{2}}{10065}$

Step 4.1.5

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 4.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\frac{- 5880 + 448 \sqrt{15} + 1575 \sqrt{15} - 120 {(15^{\frac{1}{2}})}^{2}}{10065}$

Step 4.1.5.2

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

$\frac{- 5880 + 448 \sqrt{15} + 1575 \sqrt{15} - 120 \cdot 15^{\frac{1}{2} \cdot 2}}{10065}$

Step 4.1.5.3

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

$\frac{- 5880 + 448 \sqrt{15} + 1575 \sqrt{15} - 120 \cdot 15^{\frac{2}{2}}}{10065}$

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.

$\frac{- 5880 + 448 \sqrt{15} + 1575 \sqrt{15} - 120 \cdot 15^{\frac{2}{2}}}{10065}$

Step 4.1.5.4.2

Rewrite the expression.

$\frac{- 5880 + 448 \sqrt{15} + 1575 \sqrt{15} - 120 \cdot 15^{1}}{10065}$

Step 4.1.5.5

Evaluate the exponent.

$\frac{- 5880 + 448 \sqrt{15} + 1575 \sqrt{15} - 120 \cdot 15}{10065}$

Step 4.1.6

Multiply $- 120$ by $15$ .

$\frac{- 5880 + 448 \sqrt{15} + 1575 \sqrt{15} - 1800}{10065}$

Step 4.2

Subtract $1800$ from $- 5880$ .

$\frac{- 7680 + 448 \sqrt{15} + 1575 \sqrt{15}}{10065}$

Step 4.3

Add $448 \sqrt{15}$ and $1575 \sqrt{15}$ .

$\frac{- 7680 + 2023 \sqrt{15}}{10065}$

Step 5

Rewrite $- 7680$ as $- 1 (7680)$ .

$\frac{- 1 (7680) + 2023 \sqrt{15}}{10065}$

Step 6

Factor $- 1$ out of $2023 \sqrt{15}$ .

$\frac{- 1 (7680) - (- 2023 \sqrt{15})}{10065}$

Step 7

Factor $- 1$ out of $- 1 (7680) - (- 2023 \sqrt{15})$ .

$\frac{- 1 (7680 - 2023 \sqrt{15})}{10065}$

Step 8

Move the negative in front of the fraction.

$- \frac{7680 - 2023 \sqrt{15}}{10065}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{7680 - 2023 \sqrt{15}}{10065}$

Decimal Form:

$0.01540440 \dots$