Solve 8-i/7+4i

$$\frac { 8 - i } { 7 + 4 i }$$

Evaluate

$$\frac{4}{5}-\frac{3}{5}i=0.8-0.6i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 7-4i.

$$\frac{\left(8-i\right)\left(7-4i\right)}{\left(7+4i\right)\left(7-4i\right)}$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(8-i\right)\left(7-4i\right)}{7^{2}-4^{2}i^{2}}$$

By definition, i^{2} is -1. Calculate the denominator.

$$\frac{\left(8-i\right)\left(7-4i\right)}{65}$$

Multiply complex numbers 8-i and 7-4i like you multiply binomials.

$$\frac{8\times 7+8\times \left(-4i\right)-i\times 7-\left(-4i^{2}\right)}{65}$$

By definition, i^{2} is -1.

$$\frac{8\times 7+8\times \left(-4i\right)-i\times 7-\left(-4\left(-1\right)\right)}{65}$$

Do the multiplications in 8\times 7+8\times \left(-4i\right)-i\times 7-\left(-4\left(-1\right)\right).

$$\frac{56-32i-7i-4}{65}$$

Combine the real and imaginary parts in 56-32i-7i-4.

$$\frac{56-4+\left(-32-7\right)i}{65}$$

Do the additions in 56-4+\left(-32-7\right)i.

$$\frac{52-39i}{65}$$

Divide 52-39i by 65 to get \frac{4}{5}-\frac{3}{5}i.

$$\frac{4}{5}-\frac{3}{5}i$$

Real Part

$$\frac{4}{5} = 0.8$$

Solution Steps

Multiply both numerator and denominator of \frac{8-i}{7+4i} by the complex conjugate of the denominator, 7-4i.

$$Re(\frac{\left(8-i\right)\left(7-4i\right)}{\left(7+4i\right)\left(7-4i\right)})$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$Re(\frac{\left(8-i\right)\left(7-4i\right)}{7^{2}-4^{2}i^{2}})$$

By definition, i^{2} is -1. Calculate the denominator.

$$Re(\frac{\left(8-i\right)\left(7-4i\right)}{65})$$

Multiply complex numbers 8-i and 7-4i like you multiply binomials.

$$Re(\frac{8\times 7+8\times \left(-4i\right)-i\times 7-\left(-4i^{2}\right)}{65})$$

By definition, i^{2} is -1.

$$Re(\frac{8\times 7+8\times \left(-4i\right)-i\times 7-\left(-4\left(-1\right)\right)}{65})$$

Do the multiplications in 8\times 7+8\times \left(-4i\right)-i\times 7-\left(-4\left(-1\right)\right).

$$Re(\frac{56-32i-7i-4}{65})$$

Combine the real and imaginary parts in 56-32i-7i-4.

$$Re(\frac{56-4+\left(-32-7\right)i}{65})$$

Do the additions in 56-4+\left(-32-7\right)i.

$$Re(\frac{52-39i}{65})$$

Divide 52-39i by 65 to get \frac{4}{5}-\frac{3}{5}i.

$$Re(\frac{4}{5}-\frac{3}{5}i)$$

The real part of \frac{4}{5}-\frac{3}{5}i is \frac{4}{5}.

$$\frac{4}{5}$$