Solve 5-6i/4-3i

$$\frac { 5 - 6 i } { 4 - 3 i }$$

Evaluate

$$\frac{38}{25}-\frac{9}{25}i=1.52-0.36i$$

Solution Steps

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

$$\frac{\left(5-6i\right)\left(4+3i\right)}{\left(4-3i\right)\left(4+3i\right)}$$

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

$$\frac{\left(5-6i\right)\left(4+3i\right)}{4^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(5-6i\right)\left(4+3i\right)}{25}$$

Multiply complex numbers 5-6i and 4+3i like you multiply binomials.

$$\frac{5\times 4+5\times \left(3i\right)-6i\times 4-6\times 3i^{2}}{25}$$

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

$$\frac{5\times 4+5\times \left(3i\right)-6i\times 4-6\times 3\left(-1\right)}{25}$$

Do the multiplications in 5\times 4+5\times \left(3i\right)-6i\times 4-6\times 3\left(-1\right).

$$\frac{20+15i-24i+18}{25}$$

Combine the real and imaginary parts in 20+15i-24i+18.

$$\frac{20+18+\left(15-24\right)i}{25}$$

Do the additions in 20+18+\left(15-24\right)i.

$$\frac{38-9i}{25}$$

Divide 38-9i by 25 to get \frac{38}{25}-\frac{9}{25}i.

$$\frac{38}{25}-\frac{9}{25}i$$

Real Part

$$\frac{38}{25} = 1\frac{13}{25} = 1.52$$

Solution Steps

Multiply both numerator and denominator of \frac{5-6i}{4-3i} by the complex conjugate of the denominator, 4+3i.

$$Re(\frac{\left(5-6i\right)\left(4+3i\right)}{\left(4-3i\right)\left(4+3i\right)})$$

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

$$Re(\frac{\left(5-6i\right)\left(4+3i\right)}{4^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(5-6i\right)\left(4+3i\right)}{25})$$

Multiply complex numbers 5-6i and 4+3i like you multiply binomials.

$$Re(\frac{5\times 4+5\times \left(3i\right)-6i\times 4-6\times 3i^{2}}{25})$$

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

$$Re(\frac{5\times 4+5\times \left(3i\right)-6i\times 4-6\times 3\left(-1\right)}{25})$$

Do the multiplications in 5\times 4+5\times \left(3i\right)-6i\times 4-6\times 3\left(-1\right).

$$Re(\frac{20+15i-24i+18}{25})$$

Combine the real and imaginary parts in 20+15i-24i+18.

$$Re(\frac{20+18+\left(15-24\right)i}{25})$$

Do the additions in 20+18+\left(15-24\right)i.

$$Re(\frac{38-9i}{25})$$

Divide 38-9i by 25 to get \frac{38}{25}-\frac{9}{25}i.

$$Re(\frac{38}{25}-\frac{9}{25}i)$$

The real part of \frac{38}{25}-\frac{9}{25}i is \frac{38}{25}.

$$\frac{38}{25}$$