Simplify (3-4i)/(3+4i)+(3+4i)/(3-4i)

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

Simplify each term.

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

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

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

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

Multiply $- 4$ by $3$ .

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

Multiply $3$ by $- 4$ .

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

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

Multiply $- 4$ by $- 4$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $16$ by $- 1$ .

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

Subtract $16$ from $9$ .

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

Subtract $12 i$ from $- 12 i$ .

$\frac{- 7 - 24 i}{(3 + 4 i) (3 - 4 i)} + \frac{3 + 4 i}{3 - 4 i}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 7 - 24 i}{3 (3 - 4 i) + 4 i (3 - 4 i)} + \frac{3 + 4 i}{3 - 4 i}$

Apply the distributive property.

$\frac{- 7 - 24 i}{3 \cdot 3 + 3 (- 4 i) + 4 i (3 - 4 i)} + \frac{3 + 4 i}{3 - 4 i}$

Apply the distributive property.

$\frac{- 7 - 24 i}{3 \cdot 3 + 3 (- 4 i) + 4 i \cdot 3 + 4 i (- 4 i)} + \frac{3 + 4 i}{3 - 4 i}$

Simplify.

Multiply $3$ by $3$ .

$\frac{- 7 - 24 i}{9 + 3 (- 4 i) + 4 i \cdot 3 + 4 i (- 4 i)} + \frac{3 + 4 i}{3 - 4 i}$

Multiply $- 4$ by $3$ .

$\frac{- 7 - 24 i}{9 - 12 i + 4 i \cdot 3 + 4 i (- 4 i)} + \frac{3 + 4 i}{3 - 4 i}$

Multiply $3$ by $4$ .

$\frac{- 7 - 24 i}{9 - 12 i + 12 i + 4 i (- 4 i)} + \frac{3 + 4 i}{3 - 4 i}$

Multiply $- 4$ by $4$ .

$\frac{- 7 - 24 i}{9 - 12 i + 12 i - 16 i i} + \frac{3 + 4 i}{3 - 4 i}$

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

$\frac{- 7 - 24 i}{9 - 12 i + 12 i - 16 (i^{1} i)} + \frac{3 + 4 i}{3 - 4 i}$

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

$\frac{- 7 - 24 i}{9 - 12 i + 12 i - 16 (i^{1} i^{1})} + \frac{3 + 4 i}{3 - 4 i}$

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

$\frac{- 7 - 24 i}{9 - 12 i + 12 i - 16 i^{1 + 1}} + \frac{3 + 4 i}{3 - 4 i}$

Add $1$ and $1$ .

$\frac{- 7 - 24 i}{9 - 12 i + 12 i - 16 i^{2}} + \frac{3 + 4 i}{3 - 4 i}$

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

$\frac{- 7 - 24 i}{9 + 0 - 16 i^{2}} + \frac{3 + 4 i}{3 - 4 i}$

Add $9$ and $0$ .

$\frac{- 7 - 24 i}{9 - 16 i^{2}} + \frac{3 + 4 i}{3 - 4 i}$

Simplify each term.

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

$\frac{- 7 - 24 i}{9 - 16 \cdot - 1} + \frac{3 + 4 i}{3 - 4 i}$

Multiply $- 16$ by $- 1$ .

$\frac{- 7 - 24 i}{9 + 16} + \frac{3 + 4 i}{3 - 4 i}$

Add $9$ and $16$ .

$\frac{- 7 - 24 i}{25} + \frac{3 + 4 i}{3 - 4 i}$

Split the fraction $\frac{- 7 - 24 i}{25}$ into two fractions.

$\frac{- 7}{25} + \frac{- 24 i}{25} + \frac{3 + 4 i}{3 - 4 i}$

Simplify each term.

Move the negative in front of the fraction.

$- \frac{7}{25} + \frac{- 24 i}{25} + \frac{3 + 4 i}{3 - 4 i}$

Move the negative in front of the fraction.

$- \frac{7}{25} - \frac{24 i}{25} + \frac{3 + 4 i}{3 - 4 i}$

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

$- \frac{7}{25} - \frac{24 i}{25} + \frac{3 + 4 i}{3 - 4 i} \cdot \frac{3 + 4 i}{3 + 4 i}$

Multiply.

Combine.

$- \frac{7}{25} - \frac{24 i}{25} + \frac{(3 + 4 i) (3 + 4 i)}{(3 - 4 i) (3 + 4 i)}$

Simplify the numerator.

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

Apply the distributive property.

$- \frac{7}{25} - \frac{24 i}{25} + \frac{3 (3 + 4 i) + 4 i (3 + 4 i)}{(3 - 4 i) (3 + 4 i)}$

Apply the distributive property.

$- \frac{7}{25} - \frac{24 i}{25} + \frac{3 \cdot 3 + 3 (4 i) + 4 i (3 + 4 i)}{(3 - 4 i) (3 + 4 i)}$

Apply the distributive property.

$- \frac{7}{25} - \frac{24 i}{25} + \frac{3 \cdot 3 + 3 (4 i) + 4 i \cdot 3 + 4 i (4 i)}{(3 - 4 i) (3 + 4 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{9 + 3 (4 i) + 4 i \cdot 3 + 4 i (4 i)}{(3 - 4 i) (3 + 4 i)}$

Multiply $4$ by $3$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{9 + 12 i + 4 i \cdot 3 + 4 i (4 i)}{(3 - 4 i) (3 + 4 i)}$

Multiply $3$ by $4$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{9 + 12 i + 12 i + 4 i (4 i)}{(3 - 4 i) (3 + 4 i)}$

Multiply $4 i (4 i)$ .

Multiply $4$ by $4$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{9 + 12 i + 12 i + 16 i i}{(3 - 4 i) (3 + 4 i)}$

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

$- \frac{7}{25} - \frac{24 i}{25} + \frac{9 + 12 i + 12 i + 16 (i^{1} i)}{(3 - 4 i) (3 + 4 i)}$

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

$- \frac{7}{25} - \frac{24 i}{25} + \frac{9 + 12 i + 12 i + 16 (i^{1} i^{1})}{(3 - 4 i) (3 + 4 i)}$

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

$- \frac{7}{25} - \frac{24 i}{25} + \frac{9 + 12 i + 12 i + 16 i^{1 + 1}}{(3 - 4 i) (3 + 4 i)}$

Add $1$ and $1$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{9 + 12 i + 12 i + 16 i^{2}}{(3 - 4 i) (3 + 4 i)}$

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

$- \frac{7}{25} - \frac{24 i}{25} + \frac{9 + 12 i + 12 i + 16 \cdot - 1}{(3 - 4 i) (3 + 4 i)}$

Multiply $16$ by $- 1$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{9 + 12 i + 12 i - 16}{(3 - 4 i) (3 + 4 i)}$

Subtract $16$ from $9$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 12 i + 12 i}{(3 - 4 i) (3 + 4 i)}$

Add $12 i$ and $12 i$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{(3 - 4 i) (3 + 4 i)}$

Simplify the denominator.

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

Apply the distributive property.

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{3 (3 + 4 i) - 4 i (3 + 4 i)}$

Apply the distributive property.

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{3 \cdot 3 + 3 (4 i) - 4 i (3 + 4 i)}$

Apply the distributive property.

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{3 \cdot 3 + 3 (4 i) - 4 i \cdot 3 - 4 i (4 i)}$

Simplify.

Multiply $3$ by $3$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{9 + 3 (4 i) - 4 i \cdot 3 - 4 i (4 i)}$

Multiply $4$ by $3$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{9 + 12 i - 4 i \cdot 3 - 4 i (4 i)}$

Multiply $3$ by $- 4$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{9 + 12 i - 12 i - 4 i (4 i)}$

Multiply $4$ by $- 4$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{9 + 12 i - 12 i - 16 i i}$

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

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{9 + 12 i - 12 i - 16 (i^{1} i)}$

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

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{9 + 12 i - 12 i - 16 (i^{1} i^{1})}$

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

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{9 + 12 i - 12 i - 16 i^{1 + 1}}$

Add $1$ and $1$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{9 + 12 i - 12 i - 16 i^{2}}$

Subtract $12 i$ from $12 i$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{9 + 0 - 16 i^{2}}$

Add $9$ and $0$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{9 - 16 i^{2}}$

Simplify each term.

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

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{9 - 16 \cdot - 1}$

Multiply $- 16$ by $- 1$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{9 + 16}$

Add $9$ and $16$ .

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7 + 24 i}{25}$

Split the fraction $\frac{- 7 + 24 i}{25}$ into two fractions.

$- \frac{7}{25} - \frac{24 i}{25} + \frac{- 7}{25} + \frac{24 i}{25}$

Move the negative in front of the fraction.

$- \frac{7}{25} - \frac{24 i}{25} - \frac{7}{25} + \frac{24 i}{25}$

Simplify terms.

Combine the numerators over the common denominator.

$\frac{- 7 - 24 i - 7 + 24 i}{25}$

Subtract $7$ from $- 7$ .

$\frac{- 14 - 24 i + 24 i}{25}$

Add $- 24 i$ and $24 i$ .

$\frac{- 14 + 0}{25}$

Simplify the expression.

Add $- 14$ and $0$ .

$\frac{- 14}{25}$

Move the negative in front of the fraction.

$- \frac{14}{25}$

The result can be shown in multiple forms.

Exact Form:

$- \frac{14}{25}$

Decimal Form:

$- 0.56$

Simplify (3-4i)/(3+4i)+(3+4i)/(3-4i)

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