Solve 4+6i/3-7i

$$\frac { 4 + 6 i } { 3 - 7 i }$$

Evaluate

$$-\frac{15}{29}+\frac{23}{29}i\approx -0.517241379+0.793103448i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3+7i.

$$\frac{\left(4+6i\right)\left(3+7i\right)}{\left(3-7i\right)\left(3+7i\right)}$$

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

$$\frac{\left(4+6i\right)\left(3+7i\right)}{3^{2}-7^{2}i^{2}}$$

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

$$\frac{\left(4+6i\right)\left(3+7i\right)}{58}$$

Multiply complex numbers 4+6i and 3+7i like you multiply binomials.

$$\frac{4\times 3+4\times \left(7i\right)+6i\times 3+6\times 7i^{2}}{58}$$

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

$$\frac{4\times 3+4\times \left(7i\right)+6i\times 3+6\times 7\left(-1\right)}{58}$$

Do the multiplications in 4\times 3+4\times \left(7i\right)+6i\times 3+6\times 7\left(-1\right).

$$\frac{12+28i+18i-42}{58}$$

Combine the real and imaginary parts in 12+28i+18i-42.

$$\frac{12-42+\left(28+18\right)i}{58}$$

Do the additions in 12-42+\left(28+18\right)i.

$$\frac{-30+46i}{58}$$

Divide -30+46i by 58 to get -\frac{15}{29}+\frac{23}{29}i.

$$-\frac{15}{29}+\frac{23}{29}i$$

Real Part

$$-\frac{15}{29} = -0.5172413793103449$$

Solution Steps

Multiply both numerator and denominator of \frac{4+6i}{3-7i} by the complex conjugate of the denominator, 3+7i.

$$Re(\frac{\left(4+6i\right)\left(3+7i\right)}{\left(3-7i\right)\left(3+7i\right)})$$

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

$$Re(\frac{\left(4+6i\right)\left(3+7i\right)}{3^{2}-7^{2}i^{2}})$$

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

$$Re(\frac{\left(4+6i\right)\left(3+7i\right)}{58})$$

Multiply complex numbers 4+6i and 3+7i like you multiply binomials.

$$Re(\frac{4\times 3+4\times \left(7i\right)+6i\times 3+6\times 7i^{2}}{58})$$

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

$$Re(\frac{4\times 3+4\times \left(7i\right)+6i\times 3+6\times 7\left(-1\right)}{58})$$

Do the multiplications in 4\times 3+4\times \left(7i\right)+6i\times 3+6\times 7\left(-1\right).

$$Re(\frac{12+28i+18i-42}{58})$$

Combine the real and imaginary parts in 12+28i+18i-42.

$$Re(\frac{12-42+\left(28+18\right)i}{58})$$

Do the additions in 12-42+\left(28+18\right)i.

$$Re(\frac{-30+46i}{58})$$

Divide -30+46i by 58 to get -\frac{15}{29}+\frac{23}{29}i.

$$Re(-\frac{15}{29}+\frac{23}{29}i)$$

The real part of -\frac{15}{29}+\frac{23}{29}i is -\frac{15}{29}.

$$-\frac{15}{29}$$