simplify (12i)/(3-3i) — step-by-step solution

Let's simplify:

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

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

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

\frac{12 i \times (3 + 3 i)}{3^{2} + {(-3)}^{2}}

\frac{12 i \times (3 + 3 i)}{3^{2} + {(-3)}^{2}}

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

3^{2}

3 \times 3

9

{(-3)}^{2}

(-3) \times (-3)

9

\frac{12 i \times (3 + 3 i)}{9 + 9}

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

9 + 9

18

\frac{12 i \times (3 + 3 i)}{18}

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

\frac{12 i \times (3 + 3 i)}{18}

\frac{2 i \times (3 + 3 i)}{3}

\frac{2 i \times (3 + 3 i)}{3}

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

\frac{2 i \times (3 + 3 i)}{3}

2 i \times (1 + i)

2 i \times (1 + i)

ⓘ Expand by multiplying each term inside the parentheses:

2 i \times (1 + i)

2 i + 2 i^{2}

2 i + 2 i^{2}

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

i^{2}

-1

■Multiplying by -1 gives the opposite:

2 \times (-1)

-2

2 i - 2

▶Simplified form:

- 2 + 2 i