Solve 3+2i/5-2i

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

Evaluate

$$\frac{11}{29}+\frac{16}{29}i\approx 0.379310345+0.551724138i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 5+2i.

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

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

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

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

$$\frac{\left(3+2i\right)\left(5+2i\right)}{29}$$

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

$$\frac{3\times 5+3\times \left(2i\right)+2i\times 5+2\times 2i^{2}}{29}$$

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

$$\frac{3\times 5+3\times \left(2i\right)+2i\times 5+2\times 2\left(-1\right)}{29}$$

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

$$\frac{15+6i+10i-4}{29}$$

Combine the real and imaginary parts in 15+6i+10i-4.

$$\frac{15-4+\left(6+10\right)i}{29}$$

Do the additions in 15-4+\left(6+10\right)i.

$$\frac{11+16i}{29}$$

Divide 11+16i by 29 to get \frac{11}{29}+\frac{16}{29}i.

$$\frac{11}{29}+\frac{16}{29}i$$

Real Part

$$\frac{11}{29} = 0.3793103448275862$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(3+2i\right)\left(5+2i\right)}{29})$$

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

$$Re(\frac{3\times 5+3\times \left(2i\right)+2i\times 5+2\times 2i^{2}}{29})$$

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

$$Re(\frac{3\times 5+3\times \left(2i\right)+2i\times 5+2\times 2\left(-1\right)}{29})$$

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

$$Re(\frac{15+6i+10i-4}{29})$$

Combine the real and imaginary parts in 15+6i+10i-4.

$$Re(\frac{15-4+\left(6+10\right)i}{29})$$

Do the additions in 15-4+\left(6+10\right)i.

$$Re(\frac{11+16i}{29})$$

Divide 11+16i by 29 to get \frac{11}{29}+\frac{16}{29}i.

$$Re(\frac{11}{29}+\frac{16}{29}i)$$

The real part of \frac{11}{29}+\frac{16}{29}i is \frac{11}{29}.

$$\frac{11}{29}$$