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

Let's simplify:

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

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

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

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

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

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

5^{2}

5 \times 5

25

{(-1)}^{2}

(-1) \times (-1)

1

\frac{(1 + 5 i) \times (5 + i)}{25 + 1}

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

25 + 1

26

\frac{(1 + 5 i) \times (5 + i)}{26}

ⓘ Expand by multiplying each term inside the parentheses:

(1 + 5 i) \times (5 + i)

1 \times 5

+ 1 \times i

+ 5 \times i \times 5

+ 5 \times i \times i

\frac{1 \times 5 + 1 \times i + 5 \times i \times 5 + 5 \times i \times i}{26}

ⓘ Multiplications are done before additions and subtractions:

5 \times 5

25

■Multiplying powers with the same base adds the exponents:

i \times i

i^{2}

\frac{5 + i + 25 i + 5 i^{2}}{26}

ⓘ Like terms are grouped by adding their coefficients:

i + 25 i

26 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{5 + 26 i - 5}{26}

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

5 - 5

0

\frac{0 + 26 i}{26}

ⓘ Adding 0 changes nothing:

0 + 26 i

26 i

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

\frac{26 i}{26}

i

▶Simplified form:

i