Linear Algebra Examples

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

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

Step 1

Pull out imaginary unit

i

$i$ .

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Step 1.1

Rewrite

- 324

$- 324$ as

- 1 (324)

$- 1 (324)$ .

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

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

Step 1.2

Rewrite

\sqrt{- 1 (324)}

$\sqrt{- 1 (324)}$ as

\sqrt{- 1} \cdot \sqrt{324}

$\sqrt{- 1} \cdot \sqrt{324}$ .

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

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

Step 1.3

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

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

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

Step 1.4

Rewrite

- 4

$- 4$ as

- 1 (4)

$- 1 (4)$ .

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

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

Step 1.5

Rewrite

\sqrt{- 1 (4)}

$\sqrt{- 1 (4)}$ as

\sqrt{- 1} \cdot \sqrt{4}

$\sqrt{- 1} \cdot \sqrt{4}$ .

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

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

Step 1.6

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

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

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

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

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

Step 2

Simplify the numerator.

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Step 2.1

Rewrite

324

$324$ as

18^{2}

$18^{2}$ .

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

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

Step 2.2

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

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

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

Step 2.3

Move

18

$18$ to the left of

i

$i$ .

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

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

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

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

Step 3

Simplify the denominator.

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Step 3.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 3.3

Move

2

$2$ to the left of

i

$i$ .

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

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

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

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

Step 4

Cancel the common factor of

18 + 18 i

$18 + 18 i$ and

2 + 2 i

$2 + 2 i$ .

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Step 4.1

Factor

2

$2$ out of

18

$18$ .

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

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

Step 4.2

Factor

2

$2$ out of

18 i

$18 i$ .

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

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

Step 4.3

Factor

2

$2$ out of

2 (9) + 2 (9 i)

$2 (9) + 2 (9 i)$ .

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

$2$ out of

2

$2$ .

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

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

Step 4.4.2

Factor

2

$2$ out of

2 i

$2 i$ .

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

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

Step 4.4.3

Factor

2

$2$ out of

2 (1) + 2 (i)

$2 (1) + 2 (i)$ .

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

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