Solve 25+150i/50+25i

$$\frac { 25 + 150 i } { 50 + 25 i }$$

Evaluate

$$\frac{8}{5}+\frac{11}{5}i=1.6+2.2i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 50-25i.

$$\frac{\left(25+150i\right)\left(50-25i\right)}{\left(50+25i\right)\left(50-25i\right)}$$

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

$$\frac{\left(25+150i\right)\left(50-25i\right)}{50^{2}-25^{2}i^{2}}$$

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

$$\frac{\left(25+150i\right)\left(50-25i\right)}{3125}$$

Multiply complex numbers 25+150i and 50-25i like you multiply binomials.

$$\frac{25\times 50+25\times \left(-25i\right)+150i\times 50+150\left(-25\right)i^{2}}{3125}$$

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

$$\frac{25\times 50+25\times \left(-25i\right)+150i\times 50+150\left(-25\right)\left(-1\right)}{3125}$$

Do the multiplications in 25\times 50+25\times \left(-25i\right)+150i\times 50+150\left(-25\right)\left(-1\right).

$$\frac{1250-625i+7500i+3750}{3125}$$

Combine the real and imaginary parts in 1250-625i+7500i+3750.

$$\frac{1250+3750+\left(-625+7500\right)i}{3125}$$

Do the additions in 1250+3750+\left(-625+7500\right)i.

$$\frac{5000+6875i}{3125}$$

Divide 5000+6875i by 3125 to get \frac{8}{5}+\frac{11}{5}i.

$$\frac{8}{5}+\frac{11}{5}i$$

Real Part

$$\frac{8}{5} = 1\frac{3}{5} = 1.6$$

Solution Steps

Multiply both numerator and denominator of \frac{25+150i}{50+25i} by the complex conjugate of the denominator, 50-25i.

$$Re(\frac{\left(25+150i\right)\left(50-25i\right)}{\left(50+25i\right)\left(50-25i\right)})$$

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

$$Re(\frac{\left(25+150i\right)\left(50-25i\right)}{50^{2}-25^{2}i^{2}})$$

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

$$Re(\frac{\left(25+150i\right)\left(50-25i\right)}{3125})$$

Multiply complex numbers 25+150i and 50-25i like you multiply binomials.

$$Re(\frac{25\times 50+25\times \left(-25i\right)+150i\times 50+150\left(-25\right)i^{2}}{3125})$$

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

$$Re(\frac{25\times 50+25\times \left(-25i\right)+150i\times 50+150\left(-25\right)\left(-1\right)}{3125})$$

Do the multiplications in 25\times 50+25\times \left(-25i\right)+150i\times 50+150\left(-25\right)\left(-1\right).

$$Re(\frac{1250-625i+7500i+3750}{3125})$$

Combine the real and imaginary parts in 1250-625i+7500i+3750.

$$Re(\frac{1250+3750+\left(-625+7500\right)i}{3125})$$

Do the additions in 1250+3750+\left(-625+7500\right)i.

$$Re(\frac{5000+6875i}{3125})$$

Divide 5000+6875i by 3125 to get \frac{8}{5}+\frac{11}{5}i.

$$Re(\frac{8}{5}+\frac{11}{5}i)$$

The real part of \frac{8}{5}+\frac{11}{5}i is \frac{8}{5}.

$$\frac{8}{5}$$