Algebra Examples

$\frac{\sqrt{- 4}}{(3 + i) - (2 + 3 i)}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

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

$\frac{\sqrt{- 1 (4)}}{(3 + i) - (2 + 3 i)}$

Step 1.2

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

$\frac{\sqrt{- 1} \cdot \sqrt{4}}{(3 + i) - (2 + 3 i)}$

Step 1.3

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

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

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{i \sqrt{2^{2}}}{3 + i - (2 + 3 i)}$

Step 2.2

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

$\frac{i \cdot 2}{3 + i - (2 + 3 i)}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the distributive property.

$\frac{i \cdot 2}{3 + i - 1 \cdot 2 - (3 i)}$

Step 3.2

Multiply $- 1$ by $2$ .

$\frac{i \cdot 2}{3 + i - 2 - (3 i)}$

Step 3.3

Multiply $3$ by $- 1$ .

$\frac{i \cdot 2}{3 + i - 2 - 3 i}$

Step 3.4

Subtract $2$ from $3$ .

$\frac{i \cdot 2}{1 + i - 3 i}$

Step 3.5

Subtract $3 i$ from $i$ .

$\frac{i \cdot 2}{1 - 2 i}$

Step 4

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

$\frac{2 i}{1 - 2 i}$

Step 5

Multiply the numerator and denominator of $\frac{2 i}{1 - 2 i}$ by the conjugate of $1 - 2 i$ to make the denominator real.

$\frac{2 i}{1 - 2 i} \cdot \frac{1 + 2 i}{1 + 2 i}$

Step 6

Multiply.

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Step 6.1

Combine.

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

Step 6.2

Simplify the numerator.

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Step 6.2.1

Apply the distributive property.

$\frac{2 i \cdot 1 + 2 i (2 i)}{(1 - 2 i) (1 + 2 i)}$

Step 6.2.2

Multiply $2$ by $1$ .

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

Step 6.2.3

Multiply $2 i (2 i)$ .

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Step 6.2.3.1

Multiply $2$ by $2$ .

$\frac{2 i + 4 i i}{(1 - 2 i) (1 + 2 i)}$

Step 6.2.3.2

Raise $i$ to the power of $1$ .

$\frac{2 i + 4 (i^{1} i)}{(1 - 2 i) (1 + 2 i)}$

Step 6.2.3.3

Raise $i$ to the power of $1$ .

$\frac{2 i + 4 (i^{1} i^{1})}{(1 - 2 i) (1 + 2 i)}$

Step 6.2.3.4

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

$\frac{2 i + 4 i^{1 + 1}}{(1 - 2 i) (1 + 2 i)}$

Step 6.2.3.5

Add $1$ and $1$ .

$\frac{2 i + 4 i^{2}}{(1 - 2 i) (1 + 2 i)}$

Step 6.2.4

Simplify each term.

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Step 6.2.4.1

Rewrite $i^{2}$ as $- 1$ .

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

Step 6.2.4.2

Multiply $4$ by $- 1$ .

$\frac{2 i - 4}{(1 - 2 i) (1 + 2 i)}$

Step 6.2.5

Reorder $2 i$ and $- 4$ .

$\frac{- 4 + 2 i}{(1 - 2 i) (1 + 2 i)}$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Expand $(1 - 2 i) (1 + 2 i)$ using the FOIL Method.

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Step 6.3.1.1

Apply the distributive property.

$\frac{- 4 + 2 i}{1 (1 + 2 i) - 2 i (1 + 2 i)}$

Step 6.3.1.2

Apply the distributive property.

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

Step 6.3.1.3

Apply the distributive property.

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

Step 6.3.2

Simplify.

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Step 6.3.2.1

Multiply $1$ by $1$ .

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

Step 6.3.2.2

Multiply $2$ by $1$ .

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

Step 6.3.2.3

Multiply $- 2$ by $1$ .

$\frac{- 4 + 2 i}{1 + 2 i - 2 i - 2 i (2 i)}$

Step 6.3.2.4

Multiply $2$ by $- 2$ .

$\frac{- 4 + 2 i}{1 + 2 i - 2 i - 4 i i}$

Step 6.3.2.5

Raise $i$ to the power of $1$ .

$\frac{- 4 + 2 i}{1 + 2 i - 2 i - 4 (i^{1} i)}$

Step 6.3.2.6

Raise $i$ to the power of $1$ .

$\frac{- 4 + 2 i}{1 + 2 i - 2 i - 4 (i^{1} i^{1})}$

Step 6.3.2.7

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

$\frac{- 4 + 2 i}{1 + 2 i - 2 i - 4 i^{1 + 1}}$

Step 6.3.2.8

Add $1$ and $1$ .

$\frac{- 4 + 2 i}{1 + 2 i - 2 i - 4 i^{2}}$

Step 6.3.2.9

Subtract $2 i$ from $2 i$ .

$\frac{- 4 + 2 i}{1 + 0 - 4 i^{2}}$

Step 6.3.2.10

Add $1$ and $0$ .

$\frac{- 4 + 2 i}{1 - 4 i^{2}}$

Step 6.3.3

Simplify each term.

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Step 6.3.3.1

Rewrite $i^{2}$ as $- 1$ .

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

Step 6.3.3.2

Multiply $- 4$ by $- 1$ .

$\frac{- 4 + 2 i}{1 + 4}$

Step 6.3.4

Add $1$ and $4$ .

$\frac{- 4 + 2 i}{5}$

Step 7

Split the fraction $\frac{- 4 + 2 i}{5}$ into two fractions.

$\frac{- 4}{5} + \frac{2 i}{5}$

Step 8

Move the negative in front of the fraction.

$- \frac{4}{5} + \frac{2 i}{5}$