simplify (2+7i)^2 — step-by-step solution

Let's reduce:

{(2 + 7 i)}^{2}

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

{(2 + 7 i)}^{2}

(2 + 7 i) \times (2 + 7 i)

(2 + 7 i) \times (2 + 7 i)

ⓘ Expand by multiplying each term inside the parentheses:

(2 + 7 i) \times (2 + 7 i)

2 \times 2

+ 2 \times 7 i

+ 7 i \times 2

+ 7 \times i \times 7 \times i

2 \times 2

+ 2 \times 7 i

+ 7 i \times 2

+ 7 \times i \times 7 \times i

ⓘ Multiplications are done before additions and subtractions:

2 \times 2

4

2 \times 7

14

7 \times 2

14

7 \times 7

49

■Multiplying powers with the same base adds the exponents:

i \times i

i^{2}

4

+ 14 i

+ 14 i

+ 49 i^{2}

ⓘ Like terms are grouped by adding their coefficients:

14 i + 14 i

28 i

■Multiplying by -1 gives the opposite:

49 \times (-1)

-49

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

i^{2}

-1

4 + 28 i - 49

ⓘ Start from

4

and move back

49

on the number line, going below zero:

4 - 49

-45

▶Simplified form:

- 45 + 28 i