Linear Algebra Examples

$\frac{18 + \sqrt{- 324}}{2 + \sqrt{- 4}}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

Rewrite $- 324$ as $- 1 (324)$ .

$\frac{18 + \sqrt{- 1 (324)}}{2 + \sqrt{- 4}}$

Step 1.2

Rewrite $\sqrt{- 1 (324)}$ as $\sqrt{- 1} \cdot \sqrt{324}$ .

$\frac{18 + \sqrt{- 1} \cdot \sqrt{324}}{2 + \sqrt{- 4}}$

Step 1.3

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

$\frac{18 + i \cdot \sqrt{324}}{2 + \sqrt{- 4}}$

Step 1.4

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

$\frac{18 + i \cdot \sqrt{324}}{2 + \sqrt{- 1 (4)}}$

Step 1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$\frac{18 + i \cdot \sqrt{324}}{2 + \sqrt{- 1} \cdot \sqrt{4}}$

Step 1.6

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

$\frac{18 + i \cdot \sqrt{324}}{2 + i \cdot \sqrt{4}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $324$ as $18^{2}$ .

$\frac{18 + i \cdot \sqrt{18^{2}}}{2 + i \cdot \sqrt{4}}$

Step 2.2

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

$\frac{18 + i \cdot 18}{2 + i \cdot \sqrt{4}}$

Step 2.3

Move $18$ to the left of $i$ .

$\frac{18 + 18 i}{2 + i \cdot \sqrt{4}}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{18 + 18 i}{2 + i \cdot \sqrt{2^{2}}}$

Step 3.2

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

$\frac{18 + 18 i}{2 + i \cdot 2}$

Step 3.3

Move $2$ to the left of $i$ .

$\frac{18 + 18 i}{2 + 2 i}$

Step 4

Cancel the common factor of $18 + 18 i$ and $2 + 2 i$ .

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Step 4.1

Factor $2$ out of $18$ .

$\frac{2 \cdot 9 + 18 i}{2 + 2 i}$

Step 4.2

Factor $2$ out of $18 i$ .

$\frac{2 \cdot 9 + 2 (9 i)}{2 + 2 i}$

Step 4.3

Factor $2$ out of $2 (9) + 2 (9 i)$ .

$\frac{2 (9 + 9 i)}{2 + 2 i}$

Step 4.4

Cancel the common factors.

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Step 4.4.1

Factor $2$ out of $2$ .

$\frac{2 (9 + 9 i)}{2 (1) + 2 i}$

Step 4.4.2

Factor $2$ out of $2 i$ .

$\frac{2 (9 + 9 i)}{2 (1) + 2 (i)}$

Step 4.4.3

Factor $2$ out of $2 (1) + 2 (i)$ .

$\frac{2 (9 + 9 i)}{2 (1 + i)}$

Step 4.4.4

Cancel the common factor.

$\frac{2 (9 + 9 i)}{2 (1 + i)}$

Step 4.4.5

Rewrite the expression.

$\frac{9 + 9 i}{1 + i}$