simplify (-3+6i)^2 — step-by-step solution

Let's simplify:

{(- 3 + 6 i)}^{2}

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

{(- 3 + 6 i)}^{2}

(- 3 + 6 i) \times (- 3 + 6 i)

(- 3 + 6 i) \times (- 3 + 6 i)

ⓘ Expand by multiplying each term inside the parentheses:

(- 3 + 6 i) \times (- 3 + 6 i)

- 3 \times (-3)

- 3 \times 6 i

+ 6 i \times (-3)

+ 6 \times i \times 6 \times i

- 3 \times (-3)

- 3 \times 6 i

+ 6 i \times (-3)

+ 6 \times i \times 6 \times i

ⓘ Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

-3 \times (-3)

9

■Multiplications are done before additions and subtractions:

3 \times 6

18

6 \times 6

36

■Multiplications are done before additions and subtractions. A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

6 \times (-3)

-18

■Multiplying powers with the same base adds the exponents:

i \times i

i^{2}

9

- 18 i

- 18 i

+ 36 i^{2}

ⓘ Like terms are grouped by adding their coefficients:

- 18 i - 18 i

-36 i

■Multiplying by -1 gives the opposite:

36 \times (-1)

-36

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

i^{2}

-1

9 - 36 i - 36

ⓘ Start from

9

and move back

36

on the number line, going below zero:

9 - 36

-27

▶Simplified form:

- 27 - 36 i