Basic Math Examples

\frac{4 - 7 i}{3 + 4 i}

$\frac{4 - 7 i}{3 + 4 i}$

Step 1

Multiply the numerator and denominator of

\frac{4 - 7 i}{3 + 4 i}

$\frac{4 - 7 i}{3 + 4 i}$ by the conjugate of

3 + 4 i

$3 + 4 i$ to make the denominator real.

\frac{4 - 7 i}{3 + 4 i} \cdot \frac{3 - 4 i}{3 - 4 i}

$\frac{4 - 7 i}{3 + 4 i} \cdot \frac{3 - 4 i}{3 - 4 i}$

Step 2

Multiply.

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Step 2.1

Combine.

\frac{(4 - 7 i) (3 - 4 i)}{(3 + 4 i) (3 - 4 i)}

$\frac{(4 - 7 i) (3 - 4 i)}{(3 + 4 i) (3 - 4 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(4 - 7 i) (3 - 4 i)

$(4 - 7 i) (3 - 4 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

\frac{4 (3 - 4 i) - 7 i (3 - 4 i)}{(3 + 4 i) (3 - 4 i)}

$\frac{4 (3 - 4 i) - 7 i (3 - 4 i)}{(3 + 4 i) (3 - 4 i)}$

Step 2.2.1.2

Apply the distributive property.

\frac{4 \cdot 3 + 4 (- 4 i) - 7 i (3 - 4 i)}{(3 + 4 i) (3 - 4 i)}

$\frac{4 \cdot 3 + 4 (- 4 i) - 7 i (3 - 4 i)}{(3 + 4 i) (3 - 4 i)}$

Step 2.2.1.3

Apply the distributive property.

\frac{4 \cdot 3 + 4 (- 4 i) - 7 i \cdot 3 - 7 i (- 4 i)}{(3 + 4 i) (3 - 4 i)}

$\frac{4 \cdot 3 + 4 (- 4 i) - 7 i \cdot 3 - 7 i (- 4 i)}{(3 + 4 i) (3 - 4 i)}$

\frac{4 \cdot 3 + 4 (- 4 i) - 7 i \cdot 3 - 7 i (- 4 i)}{(3 + 4 i) (3 - 4 i)}

$\frac{4 \cdot 3 + 4 (- 4 i) - 7 i \cdot 3 - 7 i (- 4 i)}{(3 + 4 i) (3 - 4 i)}$

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply

4

$4$ by

3

$3$ .

\frac{12 + 4 (- 4 i) - 7 i \cdot 3 - 7 i (- 4 i)}{(3 + 4 i) (3 - 4 i)}

$\frac{12 + 4 (- 4 i) - 7 i \cdot 3 - 7 i (- 4 i)}{(3 + 4 i) (3 - 4 i)}$

Step 2.2.2.1.2

Multiply

- 4

$- 4$ by

4

$4$ .

\frac{12 - 16 i - 7 i \cdot 3 - 7 i (- 4 i)}{(3 + 4 i) (3 - 4 i)}

$\frac{12 - 16 i - 7 i \cdot 3 - 7 i (- 4 i)}{(3 + 4 i) (3 - 4 i)}$

Step 2.2.2.1.3

Multiply

3

$3$ by

- 7

$- 7$ .

\frac{12 - 16 i - 21 i - 7 i (- 4 i)}{(3 + 4 i) (3 - 4 i)}

$\frac{12 - 16 i - 21 i - 7 i (- 4 i)}{(3 + 4 i) (3 - 4 i)}$

Step 2.2.2.1.4

Multiply

- 7 i (- 4 i)

$- 7 i (- 4 i)$ .

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Step 2.2.2.1.4.1

Multiply

- 4

$- 4$ by

- 7

$- 7$ .

\frac{12 - 16 i - 21 i + 28 i i}{(3 + 4 i) (3 - 4 i)}

$\frac{12 - 16 i - 21 i + 28 i i}{(3 + 4 i) (3 - 4 i)}$

Step 2.2.2.1.4.2

Raise

$i$ to the power of

$1$ .

$\frac{12 - 16 i - 21 i + 28 (i^{1} i)}{(3 + 4 i) (3 - 4 i)}$

Step 2.2.2.1.4.3

Raise

$i$ to the power of

$1$ .

$\frac{12 - 16 i - 21 i + 28 (i^{1} i^{1})}{(3 + 4 i) (3 - 4 i)}$

Step 2.2.2.1.4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{12 - 16 i - 21 i + 28 i^{1 + 1}}{(3 + 4 i) (3 - 4 i)}$

Step 2.2.2.1.4.5

Add

$1$ and

$1$ .

$\frac{12 - 16 i - 21 i + 28 i^{2}}{(3 + 4 i) (3 - 4 i)}$

Step 2.2.2.1.5

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{12 - 16 i - 21 i + 28 \cdot - 1}{(3 + 4 i) (3 - 4 i)}$

Step 2.2.2.1.6

Multiply

$28$ by

$- 1$ .

$\frac{12 - 16 i - 21 i - 28}{(3 + 4 i) (3 - 4 i)}$

Step 2.2.2.2

Subtract

$28$ from

$12$ .

$\frac{- 16 - 16 i - 21 i}{(3 + 4 i) (3 - 4 i)}$

Step 2.2.2.3

Subtract

$21 i$ from

$- 16 i$ .

$\frac{- 16 - 37 i}{(3 + 4 i) (3 - 4 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

$(3 + 4 i) (3 - 4 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 16 - 37 i}{3 (3 - 4 i) + 4 i (3 - 4 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 16 - 37 i}{3 \cdot 3 + 3 (- 4 i) + 4 i (3 - 4 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 16 - 37 i}{3 \cdot 3 + 3 (- 4 i) + 4 i \cdot 3 + 4 i (- 4 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

$3$ by

$3$ .

$\frac{- 16 - 37 i}{9 + 3 (- 4 i) + 4 i \cdot 3 + 4 i (- 4 i)}$

Step 2.3.2.2

Multiply

$- 4$ by

$3$ .

$\frac{- 16 - 37 i}{9 - 12 i + 4 i \cdot 3 + 4 i (- 4 i)}$

Step 2.3.2.3

Multiply

$3$ by

$4$ .

$\frac{- 16 - 37 i}{9 - 12 i + 12 i + 4 i (- 4 i)}$

Step 2.3.2.4

Multiply

$- 4$ by

$4$ .

$\frac{- 16 - 37 i}{9 - 12 i + 12 i - 16 i i}$

Step 2.3.2.5

Raise

$i$ to the power of

$1$ .

$\frac{- 16 - 37 i}{9 - 12 i + 12 i - 16 (i^{1} i)}$

Step 2.3.2.6

Raise

$i$ to the power of

$1$ .

$\frac{- 16 - 37 i}{9 - 12 i + 12 i - 16 (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 16 - 37 i}{9 - 12 i + 12 i - 16 i^{1 + 1}}$

Step 2.3.2.8

Add

$1$ and

$1$ .

$\frac{- 16 - 37 i}{9 - 12 i + 12 i - 16 i^{2}}$

Step 2.3.2.9

Add

$- 12 i$ and

$12 i$ .

$\frac{- 16 - 37 i}{9 + 0 - 16 i^{2}}$

Step 2.3.2.10

Add

$9$ and

$0$ .

$\frac{- 16 - 37 i}{9 - 16 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{- 16 - 37 i}{9 - 16 \cdot - 1}$

Step 2.3.3.2

Multiply

$- 16$ by

$- 1$ .

$\frac{- 16 - 37 i}{9 + 16}$

Step 2.3.4

Add

$9$ and

$16$ .

$\frac{- 16 - 37 i}{25}$

Step 3

Split the fraction

$\frac{- 16 - 37 i}{25}$ into two fractions.

$\frac{- 16}{25} + \frac{- 37 i}{25}$

Step 4

Simplify each term.

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Step 4.1

Move the negative in front of the fraction.

$- \frac{16}{25} + \frac{- 37 i}{25}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{16}{25} - \frac{37 i}{25}$