simplify (5+i)/i+(3-4i)/(1-i) — step-by-step solution

1) Let's reduce:

\frac{5 + i}{i} + \frac{3 - 4 i}{1 - i}

ⓘ To remove i from the denominator, multiply the numerator and the denominator by i, since i*i = -1:

\frac{5 + i}{i}

- i \times (5 + i)

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

\frac{3 - 4 i}{1 - i}

\frac{(3 - 4 i) \times (1 + i)}{1 + {(-1)}^{2}}

- i \times (5 + i) + \frac{(3 - 4 i) \times (1 + i)}{1 + {(-1)}^{2}}

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

{(-1)}^{2}

(-1) \times (-1)

1

- i \times (5 + i) + \frac{(3 - 4 i) \times (1 + i)}{1 + 1}

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

1 + 1

2

- i \times (5 + i) + \frac{(3 - 4 i) \times (1 + i)}{2}

ⓘ Expand by multiplying each term inside the parentheses:

- i \times (5 + i)

- (i \times 5 + i^{2})

(3 - 4 i) \times (1 + i)

3 \times 1

+ 3 i

- 4 i \times 1

- 4 \times i \times i

- (i \times 5 + i^{2}) + \frac{3 \times 1 + 3 i - 4 i \times 1 - 4 \times i \times i}{2}

ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (i \times 5 + i^{2})

- i \times 5 - i^{2}

■Multiplying powers with the same base adds the exponents:

i \times i

i^{2}

- i \times 5 - i^{2} + \frac{3 + 3 i - 4 i - 4 i^{2}}{2}

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

- i^{2}

- (-1)

i^{2}

-1

■Like terms are grouped by adding their coefficients:

3 i - 4 i

- i

■Multiplying by -1 gives the opposite:

- 4 \times (-1)

4

- i \times 5 - (-1) + \frac{3 - i + 4}{2}

ⓘ Subtracting a negative number is adding it, two minus signs make a plus:

- (-1)

1

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

3 + 4

7

- i \times 5 + 1 + \frac{7 - i}{2}

ⅰ

- i \times 5 + \frac{7 - i}{2}

\frac{2 \times (- i \times 5) + (7 - i)}{2}

\frac{2 \times (- i \times 5) + (7 - i)}{2} + 1

ⓘ Multiplications are done before additions and subtractions:

2 \times 5

10

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

10 \times (- i)

- 10 i

\frac{- 10 i + 7 - i}{2} + 1

ⓘ Like terms are grouped by adding their coefficients:

- 10 i - i

-11 i

\frac{- 11 i + 7}{2} + 1

ⅰ

\frac{- 11 i + 7}{2} + 1

\frac{(- 11 i + 7) + 2}{2}

\frac{(- 11 i + 7) + 2}{2}

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

7 + 2

9

\frac{- 11 i + 9}{2}

▶Simplified form:

\frac{9 - 11 i}{2}

2) The numeric value of:

\frac{9 - 11 i}{2}

is:

\frac{9}{2} - \frac{11}{2} \times i

▷

4.5 - 5.5 i