Solve 30+20i/1+5i

$$\frac { 30 + 20 i } { 1 + 5 i }$$

Evaluate

$$5-5i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 1-5i.

$$\frac{\left(30+20i\right)\left(1-5i\right)}{\left(1+5i\right)\left(1-5i\right)}$$

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

$$\frac{\left(30+20i\right)\left(1-5i\right)}{1^{2}-5^{2}i^{2}}$$

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

$$\frac{\left(30+20i\right)\left(1-5i\right)}{26}$$

Multiply complex numbers 30+20i and 1-5i like you multiply binomials.

$$\frac{30\times 1+30\times \left(-5i\right)+20i\times 1+20\left(-5\right)i^{2}}{26}$$

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

$$\frac{30\times 1+30\times \left(-5i\right)+20i\times 1+20\left(-5\right)\left(-1\right)}{26}$$

Do the multiplications in 30\times 1+30\times \left(-5i\right)+20i\times 1+20\left(-5\right)\left(-1\right).

$$\frac{30-150i+20i+100}{26}$$

Combine the real and imaginary parts in 30-150i+20i+100.

$$\frac{30+100+\left(-150+20\right)i}{26}$$

Do the additions in 30+100+\left(-150+20\right)i.

$$\frac{130-130i}{26}$$

Divide 130-130i by 26 to get 5-5i.

$$5-5i$$

Real Part

$$5$$

Solution Steps

Multiply both numerator and denominator of \frac{30+20i}{1+5i} by the complex conjugate of the denominator, 1-5i.

$$Re(\frac{\left(30+20i\right)\left(1-5i\right)}{\left(1+5i\right)\left(1-5i\right)})$$

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

$$Re(\frac{\left(30+20i\right)\left(1-5i\right)}{1^{2}-5^{2}i^{2}})$$

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

$$Re(\frac{\left(30+20i\right)\left(1-5i\right)}{26})$$

Multiply complex numbers 30+20i and 1-5i like you multiply binomials.

$$Re(\frac{30\times 1+30\times \left(-5i\right)+20i\times 1+20\left(-5\right)i^{2}}{26})$$

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

$$Re(\frac{30\times 1+30\times \left(-5i\right)+20i\times 1+20\left(-5\right)\left(-1\right)}{26})$$

Do the multiplications in 30\times 1+30\times \left(-5i\right)+20i\times 1+20\left(-5\right)\left(-1\right).

$$Re(\frac{30-150i+20i+100}{26})$$

Combine the real and imaginary parts in 30-150i+20i+100.

$$Re(\frac{30+100+\left(-150+20\right)i}{26})$$

Do the additions in 30+100+\left(-150+20\right)i.

$$Re(\frac{130-130i}{26})$$

Divide 130-130i by 26 to get 5-5i.

$$Re(5-5i)$$

The real part of 5-5i is 5.

$$5$$