Trigonometry Examples

$- 2 \sqrt{\frac{{(- 2)}^{2} - 4 \cdot 6 \cdot - 3}{2 (6)}}$

Step 1

Raise $- 2$ to the power of $2$ .

$- 2 \sqrt{\frac{4 - 4 \cdot 6 \cdot - 3}{2 (6)}}$

Step 2

Multiply $- 4 \cdot 6 \cdot - 3$ .

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Step 2.1

Multiply $- 4$ by $6$ .

$- 2 \sqrt{\frac{4 - 24 \cdot - 3}{2 (6)}}$

Step 2.2

Multiply $- 24$ by $- 3$ .

$- 2 \sqrt{\frac{4 + 72}{2 (6)}}$

Step 3

Add $4$ and $72$ .

$- 2 \sqrt{\frac{76}{2 (6)}}$

Step 4

Reduce the expression $\frac{76}{2 (6)}$ by cancelling the common factors.

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Step 4.1

Factor $2$ out of $76$ .

$- 2 \sqrt{\frac{2 \cdot 38}{2 \cdot 6}}$

Step 4.2

Factor $2$ out of $2 \cdot 6$ .

$- 2 \sqrt{\frac{2 \cdot 38}{2 (6)}}$

Step 4.3

Cancel the common factor.

$- 2 \sqrt{\frac{2 \cdot 38}{2 \cdot 6}}$

Step 4.4

Rewrite the expression.

$- 2 \sqrt{\frac{38}{6}}$

Step 5

Cancel the common factor of $38$ and $6$ .

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Step 5.1

Factor $2$ out of $38$ .

$- 2 \sqrt{\frac{2 (19)}{6}}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $2$ out of $6$ .

$- 2 \sqrt{\frac{2 \cdot 19}{2 \cdot 3}}$

Step 5.2.2

Cancel the common factor.

$- 2 \sqrt{\frac{2 \cdot 19}{2 \cdot 3}}$

Step 5.2.3

Rewrite the expression.

$- 2 \sqrt{\frac{19}{3}}$

Step 6

Rewrite $\sqrt{\frac{19}{3}}$ as $\frac{\sqrt{19}}{\sqrt{3}}$ .

$- 2 \frac{\sqrt{19}}{\sqrt{3}}$

Step 7

Multiply $\frac{\sqrt{19}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$- 2 (\frac{\sqrt{19}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{\sqrt{19}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$- 2 \frac{\sqrt{19} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 8.2

Raise $\sqrt{3}$ to the power of $1$ .

$- 2 \frac{\sqrt{19} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 8.3

Raise $\sqrt{3}$ to the power of $1$ .

$- 2 \frac{\sqrt{19} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 8.4

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

$- 2 \frac{\sqrt{19} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 8.5

Add $1$ and $1$ .

$- 2 \frac{\sqrt{19} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 8.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- 2 \frac{\sqrt{19} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 8.6.2

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

$- 2 \frac{\sqrt{19} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 8.6.3

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

$- 2 \frac{\sqrt{19} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

$- 2 \frac{\sqrt{19} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

$- 2 \frac{\sqrt{19} \sqrt{3}}{3^{1}}$

Step 8.6.5

Evaluate the exponent.

$- 2 \frac{\sqrt{19} \sqrt{3}}{3}$

Step 9

Simplify the numerator.

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Step 9.1

Combine using the product rule for radicals.

$- 2 \frac{\sqrt{19 \cdot 3}}{3}$

Step 9.2

Multiply $19$ by $3$ .

$- 2 \frac{\sqrt{57}}{3}$

Step 10

Combine fractions.

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Step 10.1

Combine $- 2$ and $\frac{\sqrt{57}}{3}$ .

$\frac{- 2 \sqrt{57}}{3}$

Step 10.2

Move the negative in front of the fraction.

$- \frac{2 \sqrt{57}}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{2 \sqrt{57}}{3}$

Decimal Form:

$- 5.03322295 \dots$