simplify (7i)/(8+i) — step-by-step solution

Let's simplify:

\frac{7 i}{8 + i}

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

\frac{7 i}{8 + i}

\frac{7 i \times (8 - i)}{8^{2} + 1}

\frac{7 i \times (8 - i)}{8^{2} + 1}

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

8^{2}

8 \times 8

64

\frac{7 i \times (8 - i)}{64 + 1}

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

64 + 1

65

\frac{7 i \times (8 - i)}{65}

ⓘ Expand by multiplying each term inside the parentheses:

7 i \times (8 - i)

56 i + 7 \times i \times (- i)

\frac{56 i + 7 \times i \times (- i)}{65}

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

7 \times (- i)

-7 i

\frac{56 i - 7 \times i \times i}{65}

ⓘ Multiplying powers with the same base adds the exponents:

i \times i

i^{2}

\frac{56 i - 7 i^{2}}{65}

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

i^{2}

-1

■Multiplying by -1 gives the opposite:

- 7 \times (-1)

7

\frac{56 i + 7}{65}

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

\frac{56 i + 7}{65}

\frac{7}{65} + \frac{56}{65} \times i

▶Simplified form:

\frac{7}{65} + \frac{56}{65} \times i

The numeric value of:

\frac{7}{65} + \frac{56}{65} \times i

equals:

▷

\approx 0.1077 + \approx 0.8615 i