Trigonometry Examples

$\frac{- 5 - 6 \sqrt{2}}{3 \sqrt{34}}$

Step 1

Multiply $\frac{- 5 - 6 \sqrt{2}}{3 \sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\frac{- 5 - 6 \sqrt{2}}{3 \sqrt{34}} \cdot \frac{\sqrt{34}}{\sqrt{34}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{- 5 - 6 \sqrt{2}}{3 \sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\frac{(- 5 - 6 \sqrt{2}) \sqrt{34}}{3 \sqrt{34} \sqrt{34}}$

Step 2.2

Move $\sqrt{34}$ .

$\frac{(- 5 - 6 \sqrt{2}) \sqrt{34}}{3 (\sqrt{34} \sqrt{34})}$

Step 2.3

Raise $\sqrt{34}$ to the power of $1$ .

$\frac{(- 5 - 6 \sqrt{2}) \sqrt{34}}{3 ({\sqrt{34}}^{1} \sqrt{34})}$

Step 2.4

Raise $\sqrt{34}$ to the power of $1$ .

$\frac{(- 5 - 6 \sqrt{2}) \sqrt{34}}{3 ({\sqrt{34}}^{1} {\sqrt{34}}^{1})}$

Step 2.5

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

$\frac{(- 5 - 6 \sqrt{2}) \sqrt{34}}{3 {\sqrt{34}}^{1 + 1}}$

Step 2.6

Add $1$ and $1$ .

$\frac{(- 5 - 6 \sqrt{2}) \sqrt{34}}{3 {\sqrt{34}}^{2}}$

Step 2.7

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

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Step 2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\frac{(- 5 - 6 \sqrt{2}) \sqrt{34}}{3 {(34^{\frac{1}{2}})}^{2}}$

Step 2.7.2

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

$\frac{(- 5 - 6 \sqrt{2}) \sqrt{34}}{3 \cdot 34^{\frac{1}{2} \cdot 2}}$

Step 2.7.3

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

$\frac{(- 5 - 6 \sqrt{2}) \sqrt{34}}{3 \cdot 34^{\frac{2}{2}}}$

Step 2.7.4

Cancel the common factor of $2$ .

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Step 2.7.4.1

Cancel the common factor.

$\frac{(- 5 - 6 \sqrt{2}) \sqrt{34}}{3 \cdot 34^{\frac{2}{2}}}$

Step 2.7.4.2

Rewrite the expression.

$\frac{(- 5 - 6 \sqrt{2}) \sqrt{34}}{3 \cdot 34^{1}}$

Step 2.7.5

Evaluate the exponent.

$\frac{(- 5 - 6 \sqrt{2}) \sqrt{34}}{3 \cdot 34}$

Step 3

Multiply $3$ by $34$ .

$\frac{(- 5 - 6 \sqrt{2}) \sqrt{34}}{102}$

Step 4

Apply the distributive property.

$\frac{- 5 \sqrt{34} - 6 \sqrt{2} \sqrt{34}}{102}$

Step 5

Multiply $- 6 \sqrt{2} \sqrt{34}$ .

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Step 5.1

Combine using the product rule for radicals.

$\frac{- 5 \sqrt{34} - 6 \sqrt{34 \cdot 2}}{102}$

Step 5.2

Multiply $34$ by $2$ .

$\frac{- 5 \sqrt{34} - 6 \sqrt{68}}{102}$

Step 6

Simplify each term.

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Step 6.1

Rewrite $68$ as $2^{2} \cdot 17$ .

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Step 6.1.1

Factor $4$ out of $68$ .

$\frac{- 5 \sqrt{34} - 6 \sqrt{4 (17)}}{102}$

Step 6.1.2

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

$\frac{- 5 \sqrt{34} - 6 \sqrt{2^{2} \cdot 17}}{102}$

Step 6.2

Pull terms out from under the radical.

$\frac{- 5 \sqrt{34} - 6 (2 \sqrt{17})}{102}$

Step 6.3

Multiply $2$ by $- 6$ .

$\frac{- 5 \sqrt{34} - 12 \sqrt{17}}{102}$

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $- 5 \sqrt{34}$ .

$\frac{- (5 \sqrt{34}) - 12 \sqrt{17}}{102}$

Step 7.2

Factor $- 1$ out of $- 12 \sqrt{17}$ .

$\frac{- (5 \sqrt{34}) - (12 \sqrt{17})}{102}$

Step 7.3

Factor $- 1$ out of $- (5 \sqrt{34}) - (12 \sqrt{17})$ .

$\frac{- (5 \sqrt{34} + 12 \sqrt{17})}{102}$

Step 7.4

Simplify the expression.

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Step 7.4.1

Rewrite $- (5 \sqrt{34} + 12 \sqrt{17})$ as $- 1 (5 \sqrt{34} + 12 \sqrt{17})$ .

$\frac{- 1 (5 \sqrt{34} + 12 \sqrt{17})}{102}$

Step 7.4.2

Move the negative in front of the fraction.

$- \frac{5 \sqrt{34} + 12 \sqrt{17}}{102}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{5 \sqrt{34} + 12 \sqrt{17}}{102}$

Decimal Form:

$- 0.77090222 \dots$