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

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Rewrite ${(2 + i)}^{2}$ as $(2 + i) (2 + i)$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$\frac{(4 + 2 i + i \cdot 2 + i i) (- i) (1 - i)}{(1 + 1 i) (1 - i)}$

Move $2$ to the left of $i$ .

$\frac{(4 + 2 i + 2 \cdot i + i i) (- i) (1 - i)}{(1 + 1 i) (1 - i)}$

Multiply $i i$ .

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

$\frac{(4 + 2 i + 2 i + i^{1} i) (- i) (1 - i)}{(1 + 1 i) (1 - i)}$

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

$\frac{(4 + 2 i + 2 i + i^{1} i^{1}) (- i) (1 - i)}{(1 + 1 i) (1 - i)}$

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

$\frac{(4 + 2 i + 2 i + i^{1 + 1}) (- i) (1 - i)}{(1 + 1 i) (1 - i)}$

Add $1$ and $1$ .

$\frac{(4 + 2 i + 2 i + i^{2}) (- i) (1 - i)}{(1 + 1 i) (1 - i)}$

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

$\frac{(4 + 2 i + 2 i - 1) (- i) (1 - i)}{(1 + 1 i) (1 - i)}$

Subtract $1$ from $4$ .

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

Add $2 i$ and $2 i$ .

$\frac{(3 + 4 i) (- i) (1 - i)}{(1 + 1 i) (1 - i)}$

Apply the distributive property.

$\frac{(3 (- i) + 4 i (- i)) (1 - i)}{(1 + 1 i) (1 - i)}$

Multiply $- 1$ by $3$ .

$\frac{(- 3 i + 4 i (- i)) (1 - i)}{(1 + 1 i) (1 - i)}$

Multiply $4 i (- i)$ .

Multiply $- 1$ by $4$ .

$\frac{(- 3 i - 4 i i) (1 - i)}{(1 + 1 i) (1 - i)}$

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

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

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

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

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

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

Add $1$ and $1$ .

$\frac{(- 3 i - 4 i^{2}) (1 - i)}{(1 + 1 i) (1 - i)}$

Simplify each term.

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

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

Multiply $- 4$ by $- 1$ .

$\frac{(- 3 i + 4) (1 - i)}{(1 + 1 i) (1 - i)}$

Expand $(- 3 i + 4) (1 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 3 i (1 - i) + 4 (1 - i)}{(1 + 1 i) (1 - i)}$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- 3$ by $1$ .

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

Multiply $- 3 i (- i)$ .

Multiply $- 1$ by $- 3$ .

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

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

$\frac{- 3 i + 3 (i^{1} i) + 4 \cdot 1 + 4 (- i)}{(1 + 1 i) (1 - i)}$

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

$\frac{- 3 i + 3 (i^{1} i^{1}) + 4 \cdot 1 + 4 (- i)}{(1 + 1 i) (1 - i)}$

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

$\frac{- 3 i + 3 i^{1 + 1} + 4 \cdot 1 + 4 (- i)}{(1 + 1 i) (1 - i)}$

Add $1$ and $1$ .

$\frac{- 3 i + 3 i^{2} + 4 \cdot 1 + 4 (- i)}{(1 + 1 i) (1 - i)}$

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

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

Multiply $3$ by $- 1$ .

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

Multiply $4$ by $1$ .

$\frac{- 3 i - 3 + 4 + 4 (- i)}{(1 + 1 i) (1 - i)}$

Multiply $- 1$ by $4$ .

$\frac{- 3 i - 3 + 4 - 4 i}{(1 + 1 i) (1 - i)}$

Subtract $4 i$ from $- 3 i$ .

$\frac{- 3 + 4 - 7 i}{(1 + 1 i) (1 - i)}$

Add $- 3$ and $4$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $1$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

Multiply $1$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

$\frac{1 - 7 i}{1 + 0 - i^{2}}$

Add $1$ and $0$ .

$\frac{1 - 7 i}{1 - i^{2}}$

Simplify each term.

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

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

Multiply $- 1$ by $- 1$ .

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

Add $1$ and $1$ .

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

Step 3

Split the fraction $\frac{1 - 7 i}{2}$ into two fractions.

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

Step 4

Move the negative in front of the fraction.

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

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

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