Algebra Examples

$\frac{- 12 + \sqrt{- 45}}{24}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

Rewrite $- 45$ as $- 1 (45)$ .

$\frac{- 12 + \sqrt{- 1 (45)}}{24}$

Step 1.2

Rewrite $\sqrt{- 1 (45)}$ as $\sqrt{- 1} \cdot \sqrt{45}$ .

$\frac{- 12 + \sqrt{- 1} \cdot \sqrt{45}}{24}$

Step 1.3

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{- 12 + i \cdot \sqrt{45}}{24}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 2.1.1

Factor $9$ out of $45$ .

$\frac{- 12 + i \cdot \sqrt{9 (5)}}{24}$

Step 2.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{- 12 + i \cdot \sqrt{3^{2} \cdot 5}}{24}$

Step 2.2

Pull terms out from under the radical.

$\frac{- 12 + i \cdot (3 \sqrt{5})}{24}$

Step 2.3

Move $3$ to the left of $i$ .

$\frac{- 12 + 3 i \sqrt{5}}{24}$

Step 3

Cancel the common factor of $- 12 + 3 i \sqrt{5}$ and $24$ .

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Step 3.1

Factor $3$ out of $- 12$ .

$\frac{3 \cdot - 4 + 3 i \sqrt{5}}{24}$

Step 3.2

Factor $3$ out of $3 i \sqrt{5}$ .

$\frac{3 \cdot - 4 + 3 (i \sqrt{5})}{24}$

Step 3.3

Factor $3$ out of $3 \cdot - 4 + 3 (i \sqrt{5})$ .

$\frac{3 \cdot (- 4 + i \sqrt{5})}{24}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $3$ out of $24$ .

$\frac{3 \cdot (- 4 + i \sqrt{5})}{3 (8)}$

Step 3.4.2

Cancel the common factor.

$\frac{3 \cdot (- 4 + i \sqrt{5})}{3 \cdot 8}$

Step 3.4.3

Rewrite the expression.

$\frac{- 4 + i \sqrt{5}}{8}$

Step 4

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{- 1 (4) + i \sqrt{5}}{8}$

Step 5

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

$\frac{- 1 (4) - (- i \sqrt{5})}{8}$

Step 6

Factor $- 1$ out of $- 1 (4) - (- i \sqrt{5})$ .

$\frac{- 1 (4 - i \sqrt{5})}{8}$

Step 7

Move the negative in front of the fraction.

$- \frac{4 - i \sqrt{5}}{8}$

Step 8

Split the fraction $\frac{4 - i \sqrt{5}}{8}$ into two fractions.

$- (\frac{4}{8} + \frac{- i \sqrt{5}}{8})$

Step 9

Cancel the common factor of $4$ and $8$ .

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Step 9.1

Factor $4$ out of $4$ .

$- (\frac{4 (1)}{8} + \frac{- i \sqrt{5}}{8})$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $4$ out of $8$ .

$- (\frac{4 \cdot 1}{4 \cdot 2} + \frac{- i \sqrt{5}}{8})$

Step 9.2.2

Cancel the common factor.

$- (\frac{4 \cdot 1}{4 \cdot 2} + \frac{- i \sqrt{5}}{8})$

Step 9.2.3

Rewrite the expression.

$- (\frac{1}{2} + \frac{- i \sqrt{5}}{8})$

Step 10

Move the negative in front of the fraction.

$- (\frac{1}{2} - \frac{i \sqrt{5}}{8})$