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

Let's reduce:

\frac{8}{5 + i}

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

\frac{8}{5 + i}

\frac{8 \times (5 - i)}{5^{2} + 1}

\frac{8 \times (5 - i)}{5^{2} + 1}

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

5^{2}

5 \times 5

25

\frac{8 \times (5 - i)}{25 + 1}

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

25 + 1

26

\frac{8 \times (5 - i)}{26}

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

\frac{8 \times (5 - i)}{26}

\frac{4 \times (5 - i)}{13}

\frac{4 \times (5 - i)}{13}

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

\frac{4 \times (5 - i)}{13}

\frac{20}{13} - \frac{4}{13} \times i

▶Simplified form:

\frac{20}{13} - \frac{4}{13} \times i

The numeric value of:

\frac{20}{13} - \frac{4}{13} \times i

is:

▷

\approx 1.5385 - \approx 0.3077 i