Write in Standard Form (3-4i)/(4+3i)

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

Step 1

Multiply the numerator and denominator of $\frac{3 - 4 i}{4 + 3 i}$ by the conjugate of $4 + 3 i$ to make the denominator real.

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $4$ .

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

Multiply $- 3$ by $3$ .

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

Multiply $4$ by $- 4$ .

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

Multiply $- 4 i (- 3 i)$ .

Multiply $- 3$ by $- 4$ .

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

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

$\frac{12 - 9 i - 16 i + 12 (i^{1} i)}{(4 + 3 i) (4 - 3 i)}$

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

$\frac{12 - 9 i - 16 i + 12 (i^{1} i^{1})}{(4 + 3 i) (4 - 3 i)}$

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

$\frac{12 - 9 i - 16 i + 12 i^{1 + 1}}{(4 + 3 i) (4 - 3 i)}$

Add $1$ and $1$ .

$\frac{12 - 9 i - 16 i + 12 i^{2}}{(4 + 3 i) (4 - 3 i)}$

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

$\frac{12 - 9 i - 16 i + 12 \cdot - 1}{(4 + 3 i) (4 - 3 i)}$

Multiply $12$ by $- 1$ .

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

Subtract $12$ from $12$ .

$\frac{0 - 9 i - 16 i}{(4 + 3 i) (4 - 3 i)}$

Subtract $9 i$ from $0$ .

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

Subtract $16 i$ from $- 9 i$ .

$\frac{- 25 i}{(4 + 3 i) (4 - 3 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 25 i}{4 (4 - 3 i) + 3 i (4 - 3 i)}$

Apply the distributive property.

$\frac{- 25 i}{4 \cdot 4 + 4 (- 3 i) + 3 i (4 - 3 i)}$

Apply the distributive property.

$\frac{- 25 i}{4 \cdot 4 + 4 (- 3 i) + 3 i \cdot 4 + 3 i (- 3 i)}$

Simplify.

Multiply $4$ by $4$ .

$\frac{- 25 i}{16 + 4 (- 3 i) + 3 i \cdot 4 + 3 i (- 3 i)}$

Multiply $- 3$ by $4$ .

$\frac{- 25 i}{16 - 12 i + 3 i \cdot 4 + 3 i (- 3 i)}$

Multiply $4$ by $3$ .

$\frac{- 25 i}{16 - 12 i + 12 i + 3 i (- 3 i)}$

Multiply $- 3$ by $3$ .

$\frac{- 25 i}{16 - 12 i + 12 i - 9 i i}$

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

$\frac{- 25 i}{16 - 12 i + 12 i - 9 (i^{1} i)}$

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

$\frac{- 25 i}{16 - 12 i + 12 i - 9 (i^{1} i^{1})}$

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

$\frac{- 25 i}{16 - 12 i + 12 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 25 i}{16 - 12 i + 12 i - 9 i^{2}}$

Add $- 12 i$ and $12 i$ .

$\frac{- 25 i}{16 + 0 - 9 i^{2}}$

Add $16$ and $0$ .

$\frac{- 25 i}{16 - 9 i^{2}}$

Simplify each term.

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

$\frac{- 25 i}{16 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$\frac{- 25 i}{16 + 9}$

Add $16$ and $9$ .

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

Step 3

Cancel the common factor of $- 25$ and $25$ .

Factor $25$ out of $- 25 i$ .

$\frac{25 (- i)}{25}$

Cancel the common factors.

Factor $25$ out of $25$ .

$\frac{25 (- i)}{25 (1)}$

Cancel the common factor.

$\frac{25 (- i)}{25 \cdot 1}$

Rewrite the expression.

$\frac{- i}{1}$

Divide $- i$ by $1$ .

$- i$

Write in Standard Form (3-4i)/(4+3i)

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