simplify (-2+i)/(6-10i) — step-by-step solution

1) Let's reduce:

\frac{- 2 + i}{6 - 10 i}

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

\frac{- 2 + i}{6 - 10 i}

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

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

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

6^{2}

6 \times 6

36

{(-10)}^{2}

(-10) \times (-10)

100

\frac{(- 2 + i) \times (6 + 10 i)}{36 + 100}

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

36 + 100

136

\frac{(- 2 + i) \times (6 + 10 i)}{136}

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

\frac{(- 2 + i) \times (6 + 10 i)}{136}

\frac{(- 2 + i) \times (3 + 5 i)}{68}

\frac{(- 2 + i) \times (3 + 5 i)}{68}

ⓘ Expand by multiplying each term inside the parentheses:

(- 2 + i) \times (3 + 5 i)

- 2 \times 3

- 2 \times 5 i

+ i \times 3

+ i \times 5 \times i

\frac{- 2 \times 3 - 2 \times 5 i + i \times 3 + i \times 5 \times i}{68}

ⓘ Multiplications are done before additions and subtractions:

2 \times 3

6

2 \times 5

10

■Multiplying powers with the same base adds the exponents:

i \times i

i^{2}

\frac{- 6 - 10 i + i \times 3 + 5 i^{2}}{68}

ⓘ Like terms are grouped by adding their coefficients:

- 10 i + i \times 3

-7 i

■Multiplying by -1 gives the opposite:

5 \times (-1)

-5

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

i^{2}

-1

\frac{- 6 - 7 i - 5}{68}

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

- 6 - 7 i - 5

- (6 + 7 i + 5)

- \frac{6 + 7 i + 5}{68}

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

6 + 5

11

- \frac{11 + 7 i}{68}

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

- \frac{11 + 7 i}{68}

- \frac{11}{68} - \frac{7}{68} \times i

▶Simplified form:

- \frac{11}{68} - \frac{7}{68} \times i

2) The numeric value of:

- \frac{11}{68} - \frac{7}{68} \times i

is:

▷

- \approx 0.1618 - \approx 0.1029 i