simplify (2+9i)/(-3-i) — step-by-step solution

1) Let's simplify:

\frac{2 + 9 i}{- 3 - i}

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

\frac{2 + 9 i}{- 3 - i}

\frac{(2 + 9 i) \times (- 3 + i)}{{(-3)}^{2} + {(-1)}^{2}}

\frac{(2 + 9 i) \times (- 3 + i)}{{(-3)}^{2} + {(-1)}^{2}}

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

{(-3)}^{2}

(-3) \times (-3)

9

{(-1)}^{2}

(-1) \times (-1)

1

\frac{(2 + 9 i) \times (- 3 + i)}{9 + 1}

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

9 + 1

10

\frac{(2 + 9 i) \times (- 3 + i)}{10}

ⓘ Expand by multiplying each term inside the parentheses:

(2 + 9 i) \times (- 3 + i)

2 \times (-3)

+ 2 i

+ 9 i \times (-3)

+ 9 \times i \times i

\frac{2 \times (-3) + 2 i + 9 i \times (-3) + 9 \times i \times i}{10}

ⓘ 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:

2 \times (-3)

-6

9 \times (-3)

-27

■Multiplying powers with the same base adds the exponents:

i \times i

i^{2}

\frac{- 6 + 2 i - 27 i + 9 i^{2}}{10}

ⓘ Like terms are grouped by adding their coefficients:

2 i - 27 i

-25 i

■Multiplying by -1 gives the opposite:

9 \times (-1)

-9

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

i^{2}

-1

\frac{- 6 - 25 i - 9}{10}

ⓘ The minus sign is factored out, changing the sign of each term:

- 6 - 25 i - 9

- (6 + 25 i + 9)

- \frac{6 + 25 i + 9}{10}

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

6 + 9

15

- \frac{15 + 25 i}{10}

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

- \frac{15 + 25 i}{10}

- \frac{3 + 5 i}{2}

- \frac{3 + 5 i}{2}

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

- \frac{3 + 5 i}{2}

- \frac{3}{2} - \frac{5}{2} \times i

▶Simplified form:

- \frac{3}{2} - \frac{5}{2} \times i

2) The numeric value of:

- \frac{3}{2} - \frac{5}{2} \times i

is:

▷

- 1.5 - 2.5 i