Solve 7-6i/2+3i=

$$\frac { 7 - 6 i } { 2 + 3 i } =$$

Evaluate

$$-\frac{4}{13}-\frac{33}{13}i\approx -0.307692308-2.538461538i$$

Solution Steps

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

$$\frac{\left(7-6i\right)\left(2-3i\right)}{\left(2+3i\right)\left(2-3i\right)}$$

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

$$\frac{\left(7-6i\right)\left(2-3i\right)}{2^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(7-6i\right)\left(2-3i\right)}{13}$$

Multiply complex numbers 7-6i and 2-3i like you multiply binomials.

$$\frac{7\times 2+7\times \left(-3i\right)-6i\times 2-6\left(-3\right)i^{2}}{13}$$

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

$$\frac{7\times 2+7\times \left(-3i\right)-6i\times 2-6\left(-3\right)\left(-1\right)}{13}$$

Do the multiplications in 7\times 2+7\times \left(-3i\right)-6i\times 2-6\left(-3\right)\left(-1\right).

$$\frac{14-21i-12i-18}{13}$$

Combine the real and imaginary parts in 14-21i-12i-18.

$$\frac{14-18+\left(-21-12\right)i}{13}$$

Do the additions in 14-18+\left(-21-12\right)i.

$$\frac{-4-33i}{13}$$

Divide -4-33i by 13 to get -\frac{4}{13}-\frac{33}{13}i.

$$-\frac{4}{13}-\frac{33}{13}i$$

Real Part

$$-\frac{4}{13} = -0.3076923076923077$$

Solution Steps

Multiply both numerator and denominator of \frac{7-6i}{2+3i} by the complex conjugate of the denominator, 2-3i.

$$Re(\frac{\left(7-6i\right)\left(2-3i\right)}{\left(2+3i\right)\left(2-3i\right)})$$

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

$$Re(\frac{\left(7-6i\right)\left(2-3i\right)}{2^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(7-6i\right)\left(2-3i\right)}{13})$$

Multiply complex numbers 7-6i and 2-3i like you multiply binomials.

$$Re(\frac{7\times 2+7\times \left(-3i\right)-6i\times 2-6\left(-3\right)i^{2}}{13})$$

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

$$Re(\frac{7\times 2+7\times \left(-3i\right)-6i\times 2-6\left(-3\right)\left(-1\right)}{13})$$

Do the multiplications in 7\times 2+7\times \left(-3i\right)-6i\times 2-6\left(-3\right)\left(-1\right).

$$Re(\frac{14-21i-12i-18}{13})$$

Combine the real and imaginary parts in 14-21i-12i-18.

$$Re(\frac{14-18+\left(-21-12\right)i}{13})$$

Do the additions in 14-18+\left(-21-12\right)i.

$$Re(\frac{-4-33i}{13})$$

Divide -4-33i by 13 to get -\frac{4}{13}-\frac{33}{13}i.

$$Re(-\frac{4}{13}-\frac{33}{13}i)$$

The real part of -\frac{4}{13}-\frac{33}{13}i is -\frac{4}{13}.

$$-\frac{4}{13}$$