Solve 4-7i/3+5i

$$\frac { 4 - 7 i } { 3 + 5 i }$$

Evaluate

$$-\frac{23}{34}-\frac{41}{34}i\approx -0.676470588-1.205882353i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3-5i.

$$\frac{\left(4-7i\right)\left(3-5i\right)}{\left(3+5i\right)\left(3-5i\right)}$$

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

$$\frac{\left(4-7i\right)\left(3-5i\right)}{3^{2}-5^{2}i^{2}}$$

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

$$\frac{\left(4-7i\right)\left(3-5i\right)}{34}$$

Multiply complex numbers 4-7i and 3-5i like you multiply binomials.

$$\frac{4\times 3+4\times \left(-5i\right)-7i\times 3-7\left(-5\right)i^{2}}{34}$$

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

$$\frac{4\times 3+4\times \left(-5i\right)-7i\times 3-7\left(-5\right)\left(-1\right)}{34}$$

Do the multiplications in 4\times 3+4\times \left(-5i\right)-7i\times 3-7\left(-5\right)\left(-1\right).

$$\frac{12-20i-21i-35}{34}$$

Combine the real and imaginary parts in 12-20i-21i-35.

$$\frac{12-35+\left(-20-21\right)i}{34}$$

Do the additions in 12-35+\left(-20-21\right)i.

$$\frac{-23-41i}{34}$$

Divide -23-41i by 34 to get -\frac{23}{34}-\frac{41}{34}i.

$$-\frac{23}{34}-\frac{41}{34}i$$

Real Part

$$-\frac{23}{34} = -0.6764705882352942$$

Solution Steps

Multiply both numerator and denominator of \frac{4-7i}{3+5i} by the complex conjugate of the denominator, 3-5i.

$$Re(\frac{\left(4-7i\right)\left(3-5i\right)}{\left(3+5i\right)\left(3-5i\right)})$$

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

$$Re(\frac{\left(4-7i\right)\left(3-5i\right)}{3^{2}-5^{2}i^{2}})$$

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

$$Re(\frac{\left(4-7i\right)\left(3-5i\right)}{34})$$

Multiply complex numbers 4-7i and 3-5i like you multiply binomials.

$$Re(\frac{4\times 3+4\times \left(-5i\right)-7i\times 3-7\left(-5\right)i^{2}}{34})$$

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

$$Re(\frac{4\times 3+4\times \left(-5i\right)-7i\times 3-7\left(-5\right)\left(-1\right)}{34})$$

Do the multiplications in 4\times 3+4\times \left(-5i\right)-7i\times 3-7\left(-5\right)\left(-1\right).

$$Re(\frac{12-20i-21i-35}{34})$$

Combine the real and imaginary parts in 12-20i-21i-35.

$$Re(\frac{12-35+\left(-20-21\right)i}{34})$$

Do the additions in 12-35+\left(-20-21\right)i.

$$Re(\frac{-23-41i}{34})$$

Divide -23-41i by 34 to get -\frac{23}{34}-\frac{41}{34}i.

$$Re(-\frac{23}{34}-\frac{41}{34}i)$$

The real part of -\frac{23}{34}-\frac{41}{34}i is -\frac{23}{34}.

$$-\frac{23}{34}$$