Simplify (2+i)/(1+i)

$\frac{2 + i}{1 + i}$

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Expand $(2 + i) (1 - i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $1$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $i$ by $1$ .

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

Multiply $i (- i)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 1$ by $- 1$ .

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

Add $2$ and $1$ .

$\frac{3 - 2 i + i}{(1 + 1 i) (1 - i)}$

Add $- 2 i$ and $i$ .

$\frac{3 - i}{(1 + 1 i) (1 - i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{3 - i}{1 (1 - i) + 1 i (1 - i)}$

Apply the distributive property.

$\frac{3 - i}{1 \cdot 1 + 1 (- i) + 1 i (1 - i)}$

Apply the distributive property.

$\frac{3 - i}{1 \cdot 1 + 1 (- i) + 1 i \cdot 1 + 1 i (- i)}$

Simplify.

Multiply $1$ by $1$ .

$\frac{3 - i}{1 + 1 (- i) + 1 i \cdot 1 + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{3 - i}{1 - 1 i + 1 i \cdot 1 + 1 i (- i)}$

Multiply $1$ by $1$ .

$\frac{3 - i}{1 - 1 i + 1 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{3 - i}{1 - 1 i + 1 i - i i}$

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

$\frac{3 - i}{1 - 1 i + 1 i - (i^{1} i)}$

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

$\frac{3 - i}{1 - 1 i + 1 i - (i^{1} i^{1})}$

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

$\frac{3 - i}{1 - 1 i + 1 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{3 - i}{1 - 1 i + 1 i - i^{2}}$

Add $- 1 i$ and $1 i$ .

$\frac{3 - i}{1 + 0 - i^{2}}$

Add $1$ and $0$ .

$\frac{3 - i}{1 - i^{2}}$

Simplify each term.

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

$\frac{3 - i}{1 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{3 - i}{1 + 1}$

Add $1$ and $1$ .

$\frac{3 - i}{2}$

Step 3

Split the fraction $\frac{3 - i}{2}$ into two fractions.

$\frac{3}{2} + \frac{- i}{2}$

Step 4

Move the negative in front of the fraction.

$\frac{3}{2} - \frac{i}{2}$

Simplify (2+i)/(1+i)

Related Questions

Support LCMGCF.com by disabling your adblocker.