simplify (-66-19i)/(-8-5i) — step-by-step solution

Let's reduce:

\frac{- 66 - 19 i}{- 8 - 5 i}

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

\frac{- 66 - 19 i}{- 8 - 5 i}

\frac{(- 66 - 19 i) \times (- 8 + 5 i)}{{(-8)}^{2} + {(-5)}^{2}}

\frac{(- 66 - 19 i) \times (- 8 + 5 i)}{{(-8)}^{2} + {(-5)}^{2}}

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

- 66 - 19 i

- (66 + 19 i)

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

{(-8)}^{2}

(-8) \times (-8)

64

{(-5)}^{2}

(-5) \times (-5)

25

\frac{- (66 + 19 i) \times (- 8 + 5 i)}{64 + 25}

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

64 + 25

89

- \frac{(66 + 19 i) \times (- 8 + 5 i)}{89}

ⓘ Expand by multiplying each term inside the parentheses:

(66 + 19 i) \times (- 8 + 5 i)

66 \times (-8)

+ 66 \times 5 i

+ 19 i \times (-8)

+ 19 \times i \times 5 \times i

- \frac{66 \times (-8) + 66 \times 5 i + 19 i \times (-8) + 19 \times i \times 5 \times i}{89}

ⓘ 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:

66 \times (-8)

- 66 \times 8

- (60 + 6) \times 8

- (60 \times 8 + 6 \times 8)

- (480 + 48)

-528

19 \times (-8)

- 19 \times 8

- (10 + 9) \times 8

- (10 \times 8 + 9 \times 8)

- (80 + 72)

-152

■Multiplications are done before additions and subtractions:

66 \times 5

(60 + 6) \times 5

60 \times 5 + 6 \times 5

300 + 30

330

19 \times 5

(10 + 9) \times 5

10 \times 5 + 9 \times 5

50 + 45

95

■Multiplying powers with the same base adds the exponents:

i \times i

i^{2}

- (\frac{- 528 + 330 i - 152 i + 95 i^{2}}{89})

ⓘ Like terms are grouped by adding their coefficients:

330 i - 152 i

178 i

■Multiplying by -1 gives the opposite:

95 \times (-1)

-95

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

i^{2}

-1

- (\frac{- 528 + 178 i - 95}{89})

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

- 528 - 95

-623

- (\frac{- 623 + 178 i}{89})

ⓘ The minus sign of the fraction goes into the numerator or the denominator, changing the sign of each of its terms:

- (\frac{- 623 + 178 i}{89})

\frac{623 - 178 i}{89}

\frac{623 - 178 i}{89}

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

\frac{623 - 178 i}{89}

\frac{89 \times (7 - 2 i)}{89}

7 - 2 i

▶Simplified form:

7 - 2 i