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

Let's reduce:

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

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

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

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

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

ⓘ Expand by multiplying each term inside the parentheses:

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

7 \times 7

+ 7 \times 2 i

+ 2 i \times 7

+ 2 \times i \times 2 \times i

7 \times 7

+ 7 \times 2 i

+ 2 i \times 7

+ 2 \times i \times 2 \times i

ⓘ Multiplications are done before additions and subtractions:

7 \times 7

49

7 \times 2

14

2 \times 7

14

2 \times 2

4

■Multiplying powers with the same base adds the exponents:

i \times i

i^{2}

49

+ 14 i

+ 14 i

+ 4 i^{2}

ⓘ Like terms are grouped by adding their coefficients:

14 i + 14 i

28 i

■Multiplying by -1 gives the opposite:

4 \times (-1)

-4

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

i^{2}

-1

49 + 28 i - 4

ⓘ Start from

49

and move back

4

on the number line:

49 - 4

45

▶Simplified form:

45 + 28 i