Trigonometry Examples

$\sqrt{{(- \frac{1}{2})}^{2} - 4 (\frac{1}{2}) (- \frac{23}{4})}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1

Apply the product rule to $- \frac{1}{2}$ .

$\sqrt{{(- 1)}^{2} {(\frac{1}{2})}^{2} - 4 (\frac{1}{2}) \cdot (- \frac{23}{4})}$

Step 1.2

Apply the product rule to $\frac{1}{2}$ .

$\sqrt{{(- 1)}^{2} \frac{1^{2}}{2^{2}} - 4 (\frac{1}{2}) \cdot (- \frac{23}{4})}$

Step 2

Simplify the expression.

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Step 2.1

Raise $- 1$ to the power of $2$ .

$\sqrt{1 \frac{1^{2}}{2^{2}} - 4 (\frac{1}{2}) \cdot (- \frac{23}{4})}$

Step 2.2

Multiply $\frac{1^{2}}{2^{2}}$ by $1$ .

$\sqrt{\frac{1^{2}}{2^{2}} - 4 (\frac{1}{2}) \cdot (- \frac{23}{4})}$

Step 2.3

One to any power is one.

$\sqrt{\frac{1}{2^{2}} - 4 (\frac{1}{2}) \cdot (- \frac{23}{4})}$

Step 2.4

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

$\sqrt{\frac{1}{4} - 4 (\frac{1}{2}) \cdot (- \frac{23}{4})}$

Step 3

Cancel the common factor of $4$ .

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Step 3.1

Move the leading negative in $- \frac{23}{4}$ into the numerator.

$\sqrt{\frac{1}{4} - 4 (\frac{1}{2}) \cdot \frac{- 23}{4}}$

Step 3.2

Factor $4$ out of $- 4 (\frac{1}{2})$ .

$\sqrt{\frac{1}{4} + 4 (- (\frac{1}{2})) \cdot \frac{- 23}{4}}$

Step 3.3

Cancel the common factor.

$\sqrt{\frac{1}{4} + 4 (- (\frac{1}{2})) \cdot \frac{- 23}{4}}$

Step 3.4

Rewrite the expression.

$\sqrt{\frac{1}{4} - (\frac{1}{2}) \cdot - 23}$

Step 4

Multiply $- 23$ by $- 1$ .

$\sqrt{\frac{1}{4} + 23 (\frac{1}{2})}$

Step 5

Combine $23$ and $\frac{1}{2}$ .

$\sqrt{\frac{1}{4} + \frac{23}{2}}$

Step 6

To write $\frac{23}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{1}{4} + \frac{23}{2} \cdot \frac{2}{2}}$

Step 7

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{23}{2}$ by $\frac{2}{2}$ .

$\sqrt{\frac{1}{4} + \frac{23 \cdot 2}{2 \cdot 2}}$

Step 7.2

Multiply $2$ by $2$ .

$\sqrt{\frac{1}{4} + \frac{23 \cdot 2}{4}}$

Step 8

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 23 \cdot 2}{4}}$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $23$ by $2$ .

$\sqrt{\frac{1 + 46}{4}}$

Step 9.2

Add $1$ and $46$ .

$\sqrt{\frac{47}{4}}$

Step 10

Rewrite $\sqrt{\frac{47}{4}}$ as $\frac{\sqrt{47}}{\sqrt{4}}$ .

$\frac{\sqrt{47}}{\sqrt{4}}$

Step 11

Simplify the denominator.

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Step 11.1

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

$\frac{\sqrt{47}}{\sqrt{2^{2}}}$

Step 11.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{47}}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{47}}{2}$

Decimal Form:

$3.42782730 \dots$