simplify (-1-i)^2 — step-by-step solution

Let's reduce:

{(- 1 - i)}^{2}

ⓘ The minus sign is factored out, changing the sign of each term:

- 1 - i

- (1 + i)

■With an even integer exponent, the minus sign in the base disappears; with an odd integer exponent, it moves in front of the power:

{(- (1 + i))}^{2}

{(1 + i)}^{2}

{(1 + i)}^{2}

ⓘ Raising a sum to an integer power is multiplying it by itself:

{(1 + i)}^{2}

(1 + i) \times (1 + i)

(1 + i) \times (1 + i)

ⓘ Expand by multiplying each term inside the parentheses:

(1 + i) \times (1 + i)

1 \times 1

+ 1 \times i

+ i \times 1

+ i \times i

1 \times 1

+ 1 \times i

+ i \times 1

+ i \times i

ⓘ Multiplications are done before additions and subtractions:

1 \times 1

1

■Multiplying powers with the same base adds the exponents:

i \times i

i^{2}

1

+ i

+ i

+ i^{2}

ⓘ Like terms are grouped by adding their coefficients:

i + i

2 i

■i^2 = -1, so the powers of i reduce to 1, i, -1 or -i:

i^{2}

-1

1 + 2 i - 1

ⓘ Start from

1

and move back

1

on the number line, down to zero:

1 - 1

0

0 + 2 i

ⓘ Adding 0 changes nothing:

0 + 2 i

2 i

▶Simplified form:

2 i