Solve (8-5i)div(3+4i)

$$( 8 - 5 i ) \div ( 3 + 4 i )$$

Evaluate

$$\frac{4}{25}-\frac{47}{25}i=0.16-1.88i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3-4i.

$$\frac{\left(8-5i\right)\left(3-4i\right)}{\left(3+4i\right)\left(3-4i\right)}$$

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

$$\frac{\left(8-5i\right)\left(3-4i\right)}{3^{2}-4^{2}i^{2}}$$

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

$$\frac{\left(8-5i\right)\left(3-4i\right)}{25}$$

Multiply complex numbers 8-5i and 3-4i like you multiply binomials.

$$\frac{8\times 3+8\times \left(-4i\right)-5i\times 3-5\left(-4\right)i^{2}}{25}$$

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

$$\frac{8\times 3+8\times \left(-4i\right)-5i\times 3-5\left(-4\right)\left(-1\right)}{25}$$

Do the multiplications in 8\times 3+8\times \left(-4i\right)-5i\times 3-5\left(-4\right)\left(-1\right).

$$\frac{24-32i-15i-20}{25}$$

Combine the real and imaginary parts in 24-32i-15i-20.

$$\frac{24-20+\left(-32-15\right)i}{25}$$

Do the additions in 24-20+\left(-32-15\right)i.

$$\frac{4-47i}{25}$$

Divide 4-47i by 25 to get \frac{4}{25}-\frac{47}{25}i.

$$\frac{4}{25}-\frac{47}{25}i$$

Real Part

$$\frac{4}{25} = 0.16$$

Solution Steps

Multiply both numerator and denominator of \frac{8-5i}{3+4i} by the complex conjugate of the denominator, 3-4i.

$$Re(\frac{\left(8-5i\right)\left(3-4i\right)}{\left(3+4i\right)\left(3-4i\right)})$$

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

$$Re(\frac{\left(8-5i\right)\left(3-4i\right)}{3^{2}-4^{2}i^{2}})$$

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

$$Re(\frac{\left(8-5i\right)\left(3-4i\right)}{25})$$

Multiply complex numbers 8-5i and 3-4i like you multiply binomials.

$$Re(\frac{8\times 3+8\times \left(-4i\right)-5i\times 3-5\left(-4\right)i^{2}}{25})$$

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

$$Re(\frac{8\times 3+8\times \left(-4i\right)-5i\times 3-5\left(-4\right)\left(-1\right)}{25})$$

Do the multiplications in 8\times 3+8\times \left(-4i\right)-5i\times 3-5\left(-4\right)\left(-1\right).

$$Re(\frac{24-32i-15i-20}{25})$$

Combine the real and imaginary parts in 24-32i-15i-20.

$$Re(\frac{24-20+\left(-32-15\right)i}{25})$$

Do the additions in 24-20+\left(-32-15\right)i.

$$Re(\frac{4-47i}{25})$$

Divide 4-47i by 25 to get \frac{4}{25}-\frac{47}{25}i.

$$Re(\frac{4}{25}-\frac{47}{25}i)$$

The real part of \frac{4}{25}-\frac{47}{25}i is \frac{4}{25}.

$$\frac{4}{25}$$