Simplify (2+i)/(5+6i)

$\frac{2 + i}{5 + 6 i}$

Multiply the numerator and denominator of $\frac{2 + i}{5 + 6 i}$ by the conjugate of $5 + 6 i$ to make the denominator real.

$\frac{2 + i}{5 + 6 i} \cdot \frac{5 - 6 i}{5 - 6 i}$

Multiply.

Combine.

$\frac{(2 + i) (5 - 6 i)}{(5 + 6 i) (5 - 6 i)}$

Simplify the numerator.

Expand $(2 + i) (5 - 6 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 (5 - 6 i) + i (5 - 6 i)}{(5 + 6 i) (5 - 6 i)}$

Apply the distributive property.

$\frac{2 \cdot 5 + 2 (- 6 i) + i (5 - 6 i)}{(5 + 6 i) (5 - 6 i)}$

Apply the distributive property.

$\frac{2 \cdot 5 + 2 (- 6 i) + i \cdot 5 + i (- 6 i)}{(5 + 6 i) (5 - 6 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $5$ .

$\frac{10 + 2 (- 6 i) + i \cdot 5 + i (- 6 i)}{(5 + 6 i) (5 - 6 i)}$

Multiply $- 6$ by $2$ .

$\frac{10 - 12 i + i \cdot 5 + i (- 6 i)}{(5 + 6 i) (5 - 6 i)}$

Move $5$ to the left of $i$ .

$\frac{10 - 12 i + 5 \cdot i + i (- 6 i)}{(5 + 6 i) (5 - 6 i)}$

Multiply $i (- 6 i)$ .

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

$\frac{10 - 12 i + 5 i - 6 (i^{1} i)}{(5 + 6 i) (5 - 6 i)}$

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

$\frac{10 - 12 i + 5 i - 6 (i^{1} i^{1})}{(5 + 6 i) (5 - 6 i)}$

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

$\frac{10 - 12 i + 5 i - 6 i^{1 + 1}}{(5 + 6 i) (5 - 6 i)}$

Add $1$ and $1$ .

$\frac{10 - 12 i + 5 i - 6 i^{2}}{(5 + 6 i) (5 - 6 i)}$

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

$\frac{10 - 12 i + 5 i - 6 \cdot - 1}{(5 + 6 i) (5 - 6 i)}$

Multiply $- 6$ by $- 1$ .

$\frac{10 - 12 i + 5 i + 6}{(5 + 6 i) (5 - 6 i)}$

Add $10$ and $6$ .

$\frac{16 - 12 i + 5 i}{(5 + 6 i) (5 - 6 i)}$

Add $- 12 i$ and $5 i$ .

$\frac{16 - 7 i}{(5 + 6 i) (5 - 6 i)}$

Simplify the denominator.

Expand $(5 + 6 i) (5 - 6 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{16 - 7 i}{5 (5 - 6 i) + 6 i (5 - 6 i)}$

Apply the distributive property.

$\frac{16 - 7 i}{5 \cdot 5 + 5 (- 6 i) + 6 i (5 - 6 i)}$

Apply the distributive property.

$\frac{16 - 7 i}{5 \cdot 5 + 5 (- 6 i) + 6 i \cdot 5 + 6 i (- 6 i)}$

Simplify.

Multiply $5$ by $5$ .

$\frac{16 - 7 i}{25 + 5 (- 6 i) + 6 i \cdot 5 + 6 i (- 6 i)}$

Multiply $- 6$ by $5$ .

$\frac{16 - 7 i}{25 - 30 i + 6 i \cdot 5 + 6 i (- 6 i)}$

Multiply $5$ by $6$ .

$\frac{16 - 7 i}{25 - 30 i + 30 i + 6 i (- 6 i)}$

Multiply $- 6$ by $6$ .

$\frac{16 - 7 i}{25 - 30 i + 30 i - 36 i i}$

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

$\frac{16 - 7 i}{25 - 30 i + 30 i - 36 (i^{1} i)}$

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

$\frac{16 - 7 i}{25 - 30 i + 30 i - 36 (i^{1} i^{1})}$

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

$\frac{16 - 7 i}{25 - 30 i + 30 i - 36 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{16 - 7 i}{25 - 30 i + 30 i - 36 i^{2}}$

Add $- 30 i$ and $30 i$ .

$\frac{16 - 7 i}{25 + 0 - 36 i^{2}}$

Add $25$ and $0$ .

$\frac{16 - 7 i}{25 - 36 i^{2}}$

Simplify each term.

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

$\frac{16 - 7 i}{25 - 36 \cdot - 1}$

Multiply $- 36$ by $- 1$ .

$\frac{16 - 7 i}{25 + 36}$

Add $25$ and $36$ .

$\frac{16 - 7 i}{61}$

Split the fraction $\frac{16 - 7 i}{61}$ into two fractions.

$\frac{16}{61} + \frac{- 7 i}{61}$

Move the negative in front of the fraction.

$\frac{16}{61} - \frac{7 i}{61}$

Simplify (2+i)/(5+6i)

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