Solve 2-36i/2+3i

$$\frac { 2 - 36 i } { 2 + 3 i }$$

Evaluate

$$-8-6i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 2-3i.

$$\frac{\left(2-36i\right)\left(2-3i\right)}{\left(2+3i\right)\left(2-3i\right)}$$

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

$$\frac{\left(2-36i\right)\left(2-3i\right)}{2^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(2-36i\right)\left(2-3i\right)}{13}$$

Multiply complex numbers 2-36i and 2-3i like you multiply binomials.

$$\frac{2\times 2+2\times \left(-3i\right)-36i\times 2-36\left(-3\right)i^{2}}{13}$$

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

$$\frac{2\times 2+2\times \left(-3i\right)-36i\times 2-36\left(-3\right)\left(-1\right)}{13}$$

Do the multiplications in 2\times 2+2\times \left(-3i\right)-36i\times 2-36\left(-3\right)\left(-1\right).

$$\frac{4-6i-72i-108}{13}$$

Combine the real and imaginary parts in 4-6i-72i-108.

$$\frac{4-108+\left(-6-72\right)i}{13}$$

Do the additions in 4-108+\left(-6-72\right)i.

$$\frac{-104-78i}{13}$$

Divide -104-78i by 13 to get -8-6i.

$$-8-6i$$

Real Part

$$-8$$

Solution Steps

Multiply both numerator and denominator of \frac{2-36i}{2+3i} by the complex conjugate of the denominator, 2-3i.

$$Re(\frac{\left(2-36i\right)\left(2-3i\right)}{\left(2+3i\right)\left(2-3i\right)})$$

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

$$Re(\frac{\left(2-36i\right)\left(2-3i\right)}{2^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(2-36i\right)\left(2-3i\right)}{13})$$

Multiply complex numbers 2-36i and 2-3i like you multiply binomials.

$$Re(\frac{2\times 2+2\times \left(-3i\right)-36i\times 2-36\left(-3\right)i^{2}}{13})$$

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

$$Re(\frac{2\times 2+2\times \left(-3i\right)-36i\times 2-36\left(-3\right)\left(-1\right)}{13})$$

Do the multiplications in 2\times 2+2\times \left(-3i\right)-36i\times 2-36\left(-3\right)\left(-1\right).

$$Re(\frac{4-6i-72i-108}{13})$$

Combine the real and imaginary parts in 4-6i-72i-108.

$$Re(\frac{4-108+\left(-6-72\right)i}{13})$$

Do the additions in 4-108+\left(-6-72\right)i.

$$Re(\frac{-104-78i}{13})$$

Divide -104-78i by 13 to get -8-6i.

$$Re(-8-6i)$$

The real part of -8-6i is -8.

$$-8$$