Evaluate 7/(7+i)

$\frac{7}{7 + i}$

Step 1

Multiply the numerator and denominator of $\frac{7}{7 + 1 i}$ by the conjugate of $7 + 1 i$ to make the denominator real.

$\frac{7}{7 + 1 i} \cdot \frac{7 - i}{7 - i}$

Step 2

Multiply.

Combine.

$\frac{7 (7 - i)}{(7 + 1 i) (7 - i)}$

Simplify the numerator.

Apply the distributive property.

$\frac{7 \cdot 7 + 7 (- i)}{(7 + 1 i) (7 - i)}$

Multiply $7$ by $7$ .

$\frac{49 + 7 (- i)}{(7 + 1 i) (7 - i)}$

Multiply $- 1$ by $7$ .

$\frac{49 - 7 i}{(7 + 1 i) (7 - i)}$

Simplify the denominator.

Expand $(7 + 1 i) (7 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{49 - 7 i}{7 (7 - i) + 1 i (7 - i)}$

Apply the distributive property.

$\frac{49 - 7 i}{7 \cdot 7 + 7 (- i) + 1 i (7 - i)}$

Apply the distributive property.

$\frac{49 - 7 i}{7 \cdot 7 + 7 (- i) + 1 i \cdot 7 + 1 i (- i)}$

Simplify.

Multiply $7$ by $7$ .

$\frac{49 - 7 i}{49 + 7 (- i) + 1 i \cdot 7 + 1 i (- i)}$

Multiply $- 1$ by $7$ .

$\frac{49 - 7 i}{49 - 7 i + 1 i \cdot 7 + 1 i (- i)}$

Multiply $7$ by $1$ .

$\frac{49 - 7 i}{49 - 7 i + 7 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{49 - 7 i}{49 - 7 i + 7 i - i i}$

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

$\frac{49 - 7 i}{49 - 7 i + 7 i - (i^{1} i)}$

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

$\frac{49 - 7 i}{49 - 7 i + 7 i - (i^{1} i^{1})}$

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

$\frac{49 - 7 i}{49 - 7 i + 7 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{49 - 7 i}{49 - 7 i + 7 i - i^{2}}$

Add $- 7 i$ and $7 i$ .

$\frac{49 - 7 i}{49 + 0 - i^{2}}$

Add $49$ and $0$ .

$\frac{49 - 7 i}{49 - i^{2}}$

Simplify each term.

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

$\frac{49 - 7 i}{49 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{49 - 7 i}{49 + 1}$

Add $49$ and $1$ .

$\frac{49 - 7 i}{50}$

Step 3

Split the fraction $\frac{49 - 7 i}{50}$ into two fractions.

$\frac{49}{50} + \frac{- 7 i}{50}$

Step 4

Move the negative in front of the fraction.

$\frac{49}{50} - \frac{7 i}{50}$

Evaluate 7/(7+i)

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