Solve 5-2i/3+4i

$$\frac { 5 - 2 i } { 3 + 4 i }$$

Evaluate

$$\frac{7}{25}-\frac{26}{25}i=0.28-1.04i$$

Solution Steps

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

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

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

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

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

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

Multiply complex numbers 5-2i and 3-4i like you multiply binomials.

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

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

$$\frac{5\times 3+5\times \left(-4i\right)-2i\times 3-2\left(-4\right)\left(-1\right)}{25}$$

Do the multiplications in 5\times 3+5\times \left(-4i\right)-2i\times 3-2\left(-4\right)\left(-1\right).

$$\frac{15-20i-6i-8}{25}$$

Combine the real and imaginary parts in 15-20i-6i-8.

$$\frac{15-8+\left(-20-6\right)i}{25}$$

Do the additions in 15-8+\left(-20-6\right)i.

$$\frac{7-26i}{25}$$

Divide 7-26i by 25 to get \frac{7}{25}-\frac{26}{25}i.

$$\frac{7}{25}-\frac{26}{25}i$$

Real Part

$$\frac{7}{25} = 0.28$$

Solution Steps

Multiply both numerator and denominator of \frac{5-2i}{3+4i} by the complex conjugate of the denominator, 3-4i.

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

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

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

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

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

Multiply complex numbers 5-2i and 3-4i like you multiply binomials.

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

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

$$Re(\frac{5\times 3+5\times \left(-4i\right)-2i\times 3-2\left(-4\right)\left(-1\right)}{25})$$

Do the multiplications in 5\times 3+5\times \left(-4i\right)-2i\times 3-2\left(-4\right)\left(-1\right).

$$Re(\frac{15-20i-6i-8}{25})$$

Combine the real and imaginary parts in 15-20i-6i-8.

$$Re(\frac{15-8+\left(-20-6\right)i}{25})$$

Do the additions in 15-8+\left(-20-6\right)i.

$$Re(\frac{7-26i}{25})$$

Divide 7-26i by 25 to get \frac{7}{25}-\frac{26}{25}i.

$$Re(\frac{7}{25}-\frac{26}{25}i)$$

The real part of \frac{7}{25}-\frac{26}{25}i is \frac{7}{25}.

$$\frac{7}{25}$$