Trigonometry Examples

$\frac{5 + \sqrt{24}}{- \frac{\sqrt{24}}{5}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(5 + \sqrt{24}) (- \frac{5}{\sqrt{24}})$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 2.1.1

Factor $4$ out of $24$ .

$(5 + \sqrt{24}) (- \frac{5}{\sqrt{4 (6)}})$

Step 2.1.2

Rewrite $4$ as $2^{2}$ .

$(5 + \sqrt{24}) (- \frac{5}{\sqrt{2^{2} \cdot 6}})$

Step 2.2

Pull terms out from under the radical.

$(5 + \sqrt{24}) (- \frac{5}{2 \sqrt{6}})$

Step 3

Simplify each term.

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Step 3.1

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 3.1.1

Factor $4$ out of $24$ .

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

Step 3.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 3.2

Pull terms out from under the radical.

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

Step 4

Multiply $\frac{5}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

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

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{5}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

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

Step 5.2

Move $\sqrt{6}$ .

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

Step 5.3

Raise $\sqrt{6}$ to the power of $1$ .

$(5 + 2 \sqrt{6}) (- \frac{5 \sqrt{6}}{2 ({\sqrt{6}}^{1} \sqrt{6})})$

Step 5.4

Raise $\sqrt{6}$ to the power of $1$ .

$(5 + 2 \sqrt{6}) (- \frac{5 \sqrt{6}}{2 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})})$

Step 5.5

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

$(5 + 2 \sqrt{6}) (- \frac{5 \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}})$

Step 5.6

Add $1$ and $1$ .

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

Step 5.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$(5 + 2 \sqrt{6}) (- \frac{5 \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}})$

Step 5.7.2

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

$(5 + 2 \sqrt{6}) (- \frac{5 \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}})$

Step 5.7.3

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

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

Step 5.7.4

Cancel the common factor of $2$ .

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Step 5.7.4.1

Cancel the common factor.

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

Step 5.7.4.2

Rewrite the expression.

$(5 + 2 \sqrt{6}) (- \frac{5 \sqrt{6}}{2 \cdot 6^{1}})$

Step 5.7.5

Evaluate the exponent.

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

Step 6

Multiply $2$ by $6$ .

$(5 + 2 \sqrt{6}) (- \frac{5 \sqrt{6}}{12})$

Step 7

Apply the distributive property.

$5 (- \frac{5 \sqrt{6}}{12}) + 2 \sqrt{6} (- \frac{5 \sqrt{6}}{12})$

Step 8

Multiply $5 (- \frac{5 \sqrt{6}}{12})$ .

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Step 8.1

Multiply $- 1$ by $5$ .

$- 5 \frac{5 \sqrt{6}}{12} + 2 \sqrt{6} (- \frac{5 \sqrt{6}}{12})$

Step 8.2

Combine $- 5$ and $\frac{5 \sqrt{6}}{12}$ .

$\frac{- 5 (5 \sqrt{6})}{12} + 2 \sqrt{6} (- \frac{5 \sqrt{6}}{12})$

Step 8.3

Multiply $5$ by $- 5$ .

$\frac{- 25 \sqrt{6}}{12} + 2 \sqrt{6} (- \frac{5 \sqrt{6}}{12})$

Step 9

Cancel the common factor of $2$ .

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Step 9.1

Move the leading negative in $- \frac{5 \sqrt{6}}{12}$ into the numerator.

$\frac{- 25 \sqrt{6}}{12} + 2 \sqrt{6} \frac{- 5 \sqrt{6}}{12}$

Step 9.2

Factor $2$ out of $2 \sqrt{6}$ .

$\frac{- 25 \sqrt{6}}{12} + 2 (\sqrt{6}) \frac{- 5 \sqrt{6}}{12}$

Step 9.3

Factor $2$ out of $12$ .

$\frac{- 25 \sqrt{6}}{12} + 2 (\sqrt{6}) \frac{- 5 \sqrt{6}}{2 (6)}$

Step 9.4

Cancel the common factor.

$\frac{- 25 \sqrt{6}}{12} + 2 \sqrt{6} \frac{- 5 \sqrt{6}}{2 \cdot 6}$

Step 9.5

Rewrite the expression.

$\frac{- 25 \sqrt{6}}{12} + \sqrt{6} \frac{- 5 \sqrt{6}}{6}$

Step 10

Combine $\sqrt{6}$ and $\frac{- 5 \sqrt{6}}{6}$ .

$\frac{- 25 \sqrt{6}}{12} + \frac{\sqrt{6} (- 5 \sqrt{6})}{6}$

Step 11

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{- 25 \sqrt{6}}{12} + \frac{- 5 ({\sqrt{6}}^{1} \sqrt{6})}{6}$

Step 12

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{- 25 \sqrt{6}}{12} + \frac{- 5 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}{6}$

Step 13

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

$\frac{- 25 \sqrt{6}}{12} + \frac{- 5 {\sqrt{6}}^{1 + 1}}{6}$

Step 14

Add $1$ and $1$ .

$\frac{- 25 \sqrt{6}}{12} + \frac{- 5 {\sqrt{6}}^{2}}{6}$

Step 15

Simplify each term.

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Step 15.1

Move the negative in front of the fraction.

$- \frac{25 \sqrt{6}}{12} + \frac{- 5 {\sqrt{6}}^{2}}{6}$

Step 15.2

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 15.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$- \frac{25 \sqrt{6}}{12} + \frac{- 5 {(6^{\frac{1}{2}})}^{2}}{6}$

Step 15.2.2

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

$- \frac{25 \sqrt{6}}{12} + \frac{- 5 \cdot 6^{\frac{1}{2} \cdot 2}}{6}$

Step 15.2.3

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

$- \frac{25 \sqrt{6}}{12} + \frac{- 5 \cdot 6^{\frac{2}{2}}}{6}$

Step 15.2.4

Cancel the common factor of $2$ .

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Step 15.2.4.1

Cancel the common factor.

$- \frac{25 \sqrt{6}}{12} + \frac{- 5 \cdot 6^{\frac{2}{2}}}{6}$

Step 15.2.4.2

Rewrite the expression.

$- \frac{25 \sqrt{6}}{12} + \frac{- 5 \cdot 6^{1}}{6}$

Step 15.2.5

Evaluate the exponent.

$- \frac{25 \sqrt{6}}{12} + \frac{- 5 \cdot 6}{6}$

Step 15.3

Multiply $- 5$ by $6$ .

$- \frac{25 \sqrt{6}}{12} + \frac{- 30}{6}$

Step 15.4

Divide $- 30$ by $6$ .

$- \frac{25 \sqrt{6}}{12} - 5$

Step 16

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$- \frac{25 \sqrt{6}}{12} - 5 \cdot \frac{12}{12}$

Step 17

Combine $- 5$ and $\frac{12}{12}$ .

$- \frac{25 \sqrt{6}}{12} + \frac{- 5 \cdot 12}{12}$

Step 18

Simplify the expression.

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Step 18.1

Combine the numerators over the common denominator.

$\frac{- 25 \sqrt{6} - 5 \cdot 12}{12}$

Step 18.2

Multiply $- 5$ by $12$ .

$\frac{- 25 \sqrt{6} - 60}{12}$

Step 19

Factor $- 1$ out of $- 25 \sqrt{6}$ .

$\frac{- (25 \sqrt{6}) - 60}{12}$

Step 20

Rewrite $- 60$ as $- 1 (60)$ .

$\frac{- (25 \sqrt{6}) - 1 (60)}{12}$

Step 21

Factor $- 1$ out of $- (25 \sqrt{6}) - 1 (60)$ .

$\frac{- (25 \sqrt{6} + 60)}{12}$

Step 22

Simplify the expression.

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Step 22.1

Rewrite $- (25 \sqrt{6} + 60)$ as $- 1 (25 \sqrt{6} + 60)$ .

$\frac{- 1 (25 \sqrt{6} + 60)}{12}$

Step 22.2

Move the negative in front of the fraction.

$- \frac{25 \sqrt{6} + 60}{12}$

Step 23

The result can be shown in multiple forms.

Exact Form:

$- \frac{25 \sqrt{6} + 60}{12}$

Decimal Form:

$- 10.10310363 \dots$