simplify (6-3i)/(4+2i) — step-by-step solution

1) Let's simplify:

\frac{6 - 3 i}{4 + 2 i}

ⓘ To make the denominator real, multiply the numerator and denominator by its conjugate, since (a+bi)(a-bi) = a²+b²:

\frac{6 - 3 i}{4 + 2 i}

\frac{(6 - 3 i) \times (4 - 2 i)}{4^{2} + 2^{2}}

\frac{(6 - 3 i) \times (4 - 2 i)}{4^{2} + 2^{2}}

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

4^{2}

4 \times 4

16

2^{2}

2 \times 2

4

\frac{(6 - 3 i) \times (4 - 2 i)}{16 + 4}

ⓘ The numerator and the denominator are computed before the division:

16 + 4

20

\frac{(6 - 3 i) \times (4 - 2 i)}{20}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{(6 - 3 i) \times (4 - 2 i)}{20}

\frac{(6 - 3 i) \times (2 - i)}{10}

\frac{(6 - 3 i) \times (2 - i)}{10}

ⓘ Expand by multiplying each term inside the parentheses:

(6 - 3 i) \times (2 - i)

6 \times 2

+ 6 \times (- i)

- 3 i \times 2

- 3 \times i \times (- i)

\frac{6 \times 2 + 6 \times (- i) - 3 i \times 2 - 3 \times i \times (- i)}{10}

ⓘ Multiplications are done before additions and subtractions:

6 \times 2

12

3 \times 2

6

■The minus sign is placed in front of the product:

6 \times (- i)

- 6 i

■Two minus signs cancel out:

-3 \times (- i)

3 i

\frac{12 - 6 i - 6 i + 3 \times i \times i}{10}

ⓘ Like terms are grouped by adding their coefficients:

- 6 i - 6 i

-12 i

■Multiplying powers with the same base adds the exponents:

i \times i

i^{2}

\frac{12 - 12 i + 3 i^{2}}{10}

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

i^{2}

-1

■Multiplying by -1 gives the opposite:

3 \times (-1)

-3

\frac{12 - 12 i - 3}{10}

ⓘ The numerator and the denominator are computed before the division:

12 - 3

9

\frac{9 - 12 i}{10}

ⓘ A complex number is written in algebraic form a + b*i, a being its real part and b its imaginary part:

\frac{9 - 12 i}{10}

\frac{9}{10} - \frac{6}{5} \times i

▶Simplified form:

\frac{9}{10} - \frac{6}{5} \times i

2) The numeric value of:

\frac{9}{10} - \frac{6}{5} \times i

equals:

▷

0.9 - 1.2 i