Basic Math Examples

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

Step 1

Cancel the common factor of $1 - 2 i$ .

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Step 1.1

Cancel the common factor.

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

Step 1.2

Rewrite the expression.

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

Step 2

Multiply $\frac{3 - 2 i}{1 + 2 i}$ by $1$ .

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

Step 3

Multiply the numerator and denominator of $\frac{3 - 2 i}{1 + 2 i}$ by the conjugate of $1 + 2 i$ to make the denominator real.

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

Step 4

Multiply.

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Step 4.1

Combine.

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

Step 4.2

Simplify the numerator.

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Step 4.2.1

Expand $(3 - 2 i) (1 - 2 i)$ using the FOIL Method.

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Step 4.2.1.1

Apply the distributive property.

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

Step 4.2.1.2

Apply the distributive property.

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

Step 4.2.1.3

Apply the distributive property.

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

Step 4.2.2

Simplify and combine like terms.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Multiply $3$ by $1$ .

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

Step 4.2.2.1.2

Multiply $- 2$ by $3$ .

$\frac{3 - 6 i - 2 i \cdot 1 - 2 i (- 2 i)}{(1 + 2 i) (1 - 2 i)}$

Step 4.2.2.1.3

Multiply $- 2$ by $1$ .

$\frac{3 - 6 i - 2 i - 2 i (- 2 i)}{(1 + 2 i) (1 - 2 i)}$

Step 4.2.2.1.4

Multiply $- 2 i (- 2 i)$ .

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Step 4.2.2.1.4.1

Multiply $- 2$ by $- 2$ .

$\frac{3 - 6 i - 2 i + 4 i i}{(1 + 2 i) (1 - 2 i)}$

Step 4.2.2.1.4.2

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

$\frac{3 - 6 i - 2 i + 4 (i^{1} i)}{(1 + 2 i) (1 - 2 i)}$

Step 4.2.2.1.4.3

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

$\frac{3 - 6 i - 2 i + 4 (i^{1} i^{1})}{(1 + 2 i) (1 - 2 i)}$

Step 4.2.2.1.4.4

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

$\frac{3 - 6 i - 2 i + 4 i^{1 + 1}}{(1 + 2 i) (1 - 2 i)}$

Step 4.2.2.1.4.5

Add $1$ and $1$ .

$\frac{3 - 6 i - 2 i + 4 i^{2}}{(1 + 2 i) (1 - 2 i)}$

Step 4.2.2.1.5

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

$\frac{3 - 6 i - 2 i + 4 \cdot - 1}{(1 + 2 i) (1 - 2 i)}$

Step 4.2.2.1.6

Multiply $4$ by $- 1$ .

$\frac{3 - 6 i - 2 i - 4}{(1 + 2 i) (1 - 2 i)}$

Step 4.2.2.2

Subtract $4$ from $3$ .

$\frac{- 1 - 6 i - 2 i}{(1 + 2 i) (1 - 2 i)}$

Step 4.2.2.3

Subtract $2 i$ from $- 6 i$ .

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

Step 4.3

Simplify the denominator.

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Step 4.3.1

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

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Step 4.3.1.1

Apply the distributive property.

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

Step 4.3.1.2

Apply the distributive property.

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

Step 4.3.1.3

Apply the distributive property.

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

Step 4.3.2

Simplify.

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Step 4.3.2.1

Multiply $1$ by $1$ .

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

Step 4.3.2.2

Multiply $- 2$ by $1$ .

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

Step 4.3.2.3

Multiply $2$ by $1$ .

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

Step 4.3.2.4

Multiply $- 2$ by $2$ .

$\frac{- 1 - 8 i}{1 - 2 i + 2 i - 4 i i}$

Step 4.3.2.5

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

$\frac{- 1 - 8 i}{1 - 2 i + 2 i - 4 (i^{1} i)}$

Step 4.3.2.6

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

$\frac{- 1 - 8 i}{1 - 2 i + 2 i - 4 (i^{1} i^{1})}$

Step 4.3.2.7

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

$\frac{- 1 - 8 i}{1 - 2 i + 2 i - 4 i^{1 + 1}}$

Step 4.3.2.8

Add $1$ and $1$ .

$\frac{- 1 - 8 i}{1 - 2 i + 2 i - 4 i^{2}}$

Step 4.3.2.9

Add $- 2 i$ and $2 i$ .

$\frac{- 1 - 8 i}{1 + 0 - 4 i^{2}}$

Step 4.3.2.10

Add $1$ and $0$ .

$\frac{- 1 - 8 i}{1 - 4 i^{2}}$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

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

$\frac{- 1 - 8 i}{1 - 4 \cdot - 1}$

Step 4.3.3.2

Multiply $- 4$ by $- 1$ .

$\frac{- 1 - 8 i}{1 + 4}$

Step 4.3.4

Add $1$ and $4$ .

$\frac{- 1 - 8 i}{5}$

Step 5

Split the fraction $\frac{- 1 - 8 i}{5}$ into two fractions.

$\frac{- 1}{5} + \frac{- 8 i}{5}$

Step 6

Simplify each term.

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Step 6.1

Move the negative in front of the fraction.

$- \frac{1}{5} + \frac{- 8 i}{5}$

Step 6.2

Move the negative in front of the fraction.

$- \frac{1}{5} - \frac{8 i}{5}$