Solve 4+6i/5-2i

$$\frac{ 4+6i }{ 5-2i }$$

Evaluate

$$\frac{8}{29}+\frac{38}{29}i\approx 0.275862069+1.310344828i$$

Solution Steps

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

$$\frac{\left(4+6i\right)\left(5+2i\right)}{\left(5-2i\right)\left(5+2i\right)}$$

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

$$\frac{\left(4+6i\right)\left(5+2i\right)}{5^{2}-2^{2}i^{2}}$$

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

$$\frac{\left(4+6i\right)\left(5+2i\right)}{29}$$

Multiply complex numbers 4+6i and 5+2i like you multiply binomials.

$$\frac{4\times 5+4\times \left(2i\right)+6i\times 5+6\times 2i^{2}}{29}$$

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

$$\frac{4\times 5+4\times \left(2i\right)+6i\times 5+6\times 2\left(-1\right)}{29}$$

Do the multiplications in 4\times 5+4\times \left(2i\right)+6i\times 5+6\times 2\left(-1\right).

$$\frac{20+8i+30i-12}{29}$$

Combine the real and imaginary parts in 20+8i+30i-12.

$$\frac{20-12+\left(8+30\right)i}{29}$$

Do the additions in 20-12+\left(8+30\right)i.

$$\frac{8+38i}{29}$$

Divide 8+38i by 29 to get \frac{8}{29}+\frac{38}{29}i.

$$\frac{8}{29}+\frac{38}{29}i$$

Real Part

$$\frac{8}{29} = 0.27586206896551724$$

Solution Steps

Multiply both numerator and denominator of \frac{4+6i}{5-2i} by the complex conjugate of the denominator, 5+2i.

$$Re(\frac{\left(4+6i\right)\left(5+2i\right)}{\left(5-2i\right)\left(5+2i\right)})$$

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

$$Re(\frac{\left(4+6i\right)\left(5+2i\right)}{5^{2}-2^{2}i^{2}})$$

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

$$Re(\frac{\left(4+6i\right)\left(5+2i\right)}{29})$$

Multiply complex numbers 4+6i and 5+2i like you multiply binomials.

$$Re(\frac{4\times 5+4\times \left(2i\right)+6i\times 5+6\times 2i^{2}}{29})$$

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

$$Re(\frac{4\times 5+4\times \left(2i\right)+6i\times 5+6\times 2\left(-1\right)}{29})$$

Do the multiplications in 4\times 5+4\times \left(2i\right)+6i\times 5+6\times 2\left(-1\right).

$$Re(\frac{20+8i+30i-12}{29})$$

Combine the real and imaginary parts in 20+8i+30i-12.

$$Re(\frac{20-12+\left(8+30\right)i}{29})$$

Do the additions in 20-12+\left(8+30\right)i.

$$Re(\frac{8+38i}{29})$$

Divide 8+38i by 29 to get \frac{8}{29}+\frac{38}{29}i.

$$Re(\frac{8}{29}+\frac{38}{29}i)$$

The real part of \frac{8}{29}+\frac{38}{29}i is \frac{8}{29}.

$$\frac{8}{29}$$