Solve 40+20i/2,5+7,5i

$$\frac { 40 + 20 i } { 2,5 + 7,5 i }$$

Evaluate

$$4-4i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 2.5-7.5i.

$$\frac{\left(40+20i\right)\left(2.5-7.5i\right)}{\left(2.5+7.5i\right)\left(2.5-7.5i\right)}$$

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

$$\frac{\left(40+20i\right)\left(2.5-7.5i\right)}{2.5^{2}-7.5^{2}i^{2}}$$

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

$$\frac{\left(40+20i\right)\left(2.5-7.5i\right)}{62.5}$$

Multiply complex numbers 40+20i and 2.5-7.5i like you multiply binomials.

$$\frac{40\times 2.5+40\times \left(-7.5i\right)+20i\times 2.5+20\left(-7.5\right)i^{2}}{62.5}$$

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

$$\frac{40\times 2.5+40\times \left(-7.5i\right)+20i\times 2.5+20\left(-7.5\right)\left(-1\right)}{62.5}$$

Do the multiplications in 40\times 2.5+40\times \left(-7.5i\right)+20i\times 2.5+20\left(-7.5\right)\left(-1\right).

$$\frac{100-300i+50i+150}{62.5}$$

Combine the real and imaginary parts in 100-300i+50i+150.

$$\frac{100+150+\left(-300+50\right)i}{62.5}$$

Do the additions in 100+150+\left(-300+50\right)i.

$$\frac{250-250i}{62.5}$$

Divide 250-250i by 62.5 to get 4-4i.

$$4-4i$$

Real Part

$$4$$

Solution Steps

Multiply both numerator and denominator of \frac{40+20i}{2.5+7.5i} by the complex conjugate of the denominator, 2.5-7.5i.

$$Re(\frac{\left(40+20i\right)\left(2.5-7.5i\right)}{\left(2.5+7.5i\right)\left(2.5-7.5i\right)})$$

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

$$Re(\frac{\left(40+20i\right)\left(2.5-7.5i\right)}{2.5^{2}-7.5^{2}i^{2}})$$

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

$$Re(\frac{\left(40+20i\right)\left(2.5-7.5i\right)}{62.5})$$

Multiply complex numbers 40+20i and 2.5-7.5i like you multiply binomials.

$$Re(\frac{40\times 2.5+40\times \left(-7.5i\right)+20i\times 2.5+20\left(-7.5\right)i^{2}}{62.5})$$

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

$$Re(\frac{40\times 2.5+40\times \left(-7.5i\right)+20i\times 2.5+20\left(-7.5\right)\left(-1\right)}{62.5})$$

Do the multiplications in 40\times 2.5+40\times \left(-7.5i\right)+20i\times 2.5+20\left(-7.5\right)\left(-1\right).

$$Re(\frac{100-300i+50i+150}{62.5})$$

Combine the real and imaginary parts in 100-300i+50i+150.

$$Re(\frac{100+150+\left(-300+50\right)i}{62.5})$$

Do the additions in 100+150+\left(-300+50\right)i.

$$Re(\frac{250-250i}{62.5})$$

Divide 250-250i by 62.5 to get 4-4i.

$$Re(4-4i)$$

The real part of 4-4i is 4.

$$4$$