Simplify (2+i)/(7+i)

$\frac{2 + i}{7 + i}$

Step 1

Multiply the numerator and denominator of $\frac{2 + i}{7 + 1 i}$ by the conjugate of $7 + 1 i$ to make the denominator real.

$\frac{2 + i}{7 + 1 i} \cdot \frac{7 - i}{7 - i}$

Step 2

Multiply.

Combine.

$\frac{(2 + i) (7 - i)}{(7 + 1 i) (7 - i)}$

Simplify the numerator.

Expand $(2 + i) (7 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 (7 - i) + i (7 - i)}{(7 + 1 i) (7 - i)}$

Apply the distributive property.

$\frac{2 \cdot 7 + 2 (- i) + i (7 - i)}{(7 + 1 i) (7 - i)}$

Apply the distributive property.

$\frac{2 \cdot 7 + 2 (- i) + i \cdot 7 + i (- i)}{(7 + 1 i) (7 - i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $7$ .

$\frac{14 + 2 (- i) + i \cdot 7 + i (- i)}{(7 + 1 i) (7 - i)}$

Multiply $- 1$ by $2$ .

$\frac{14 - 2 i + i \cdot 7 + i (- i)}{(7 + 1 i) (7 - i)}$

Move $7$ to the left of $i$ .

$\frac{14 - 2 i + 7 \cdot i + i (- i)}{(7 + 1 i) (7 - i)}$

Multiply $i (- i)$ .

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

$\frac{14 - 2 i + 7 i - (i^{1} i)}{(7 + 1 i) (7 - i)}$

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

$\frac{14 - 2 i + 7 i - (i^{1} i^{1})}{(7 + 1 i) (7 - i)}$

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

$\frac{14 - 2 i + 7 i - i^{1 + 1}}{(7 + 1 i) (7 - i)}$

Add $1$ and $1$ .

$\frac{14 - 2 i + 7 i - i^{2}}{(7 + 1 i) (7 - i)}$

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

$\frac{14 - 2 i + 7 i - - 1}{(7 + 1 i) (7 - i)}$

Multiply $- 1$ by $- 1$ .

$\frac{14 - 2 i + 7 i + 1}{(7 + 1 i) (7 - i)}$

Add $14$ and $1$ .

$\frac{15 - 2 i + 7 i}{(7 + 1 i) (7 - i)}$

Add $- 2 i$ and $7 i$ .

$\frac{15 + 5 i}{(7 + 1 i) (7 - i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{15 + 5 i}{7 (7 - i) + 1 i (7 - i)}$

Apply the distributive property.

$\frac{15 + 5 i}{7 \cdot 7 + 7 (- i) + 1 i (7 - i)}$

Apply the distributive property.

$\frac{15 + 5 i}{7 \cdot 7 + 7 (- i) + 1 i \cdot 7 + 1 i (- i)}$

Simplify.

Multiply $7$ by $7$ .

$\frac{15 + 5 i}{49 + 7 (- i) + 1 i \cdot 7 + 1 i (- i)}$

Multiply $- 1$ by $7$ .

$\frac{15 + 5 i}{49 - 7 i + 1 i \cdot 7 + 1 i (- i)}$

Multiply $7$ by $1$ .

$\frac{15 + 5 i}{49 - 7 i + 7 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{15 + 5 i}{49 - 7 i + 7 i - i i}$

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

$\frac{15 + 5 i}{49 - 7 i + 7 i - (i^{1} i)}$

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

$\frac{15 + 5 i}{49 - 7 i + 7 i - (i^{1} i^{1})}$

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

$\frac{15 + 5 i}{49 - 7 i + 7 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{15 + 5 i}{49 - 7 i + 7 i - i^{2}}$

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

$\frac{15 + 5 i}{49 + 0 - i^{2}}$

Add $49$ and $0$ .

$\frac{15 + 5 i}{49 - i^{2}}$

Simplify each term.

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

$\frac{15 + 5 i}{49 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{15 + 5 i}{49 + 1}$

Add $49$ and $1$ .

$\frac{15 + 5 i}{50}$

Step 3

Cancel the common factor of $15 + 5 i$ and $50$ .

Factor $5$ out of $15$ .

$\frac{5 \cdot 3 + 5 i}{50}$

Factor $5$ out of $5 i$ .

$\frac{5 \cdot 3 + 5 (i)}{50}$

Factor $5$ out of $5 \cdot 3 + 5 (i)$ .

$\frac{5 \cdot (3 + i)}{50}$

Cancel the common factors.

Factor $5$ out of $50$ .

$\frac{5 \cdot (3 + i)}{5 (10)}$

Cancel the common factor.

$\frac{5 \cdot (3 + i)}{5 \cdot 10}$

Rewrite the expression.

$\frac{3 + i}{10}$

Step 4

Split the fraction $\frac{3 + i}{10}$ into two fractions.

$\frac{3}{10} + \frac{i}{10}$

Simplify (2+i)/(7+i)

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