Solve (4+4i)div(5+4i)=

$$( 4 + 4 i ) \div ( 5 + 4 i ) =$$

Evaluate

$$\frac{36}{41}+\frac{4}{41}i\approx 0.87804878+0.097560976i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 5-4i.

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

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

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

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

$$\frac{\left(4+4i\right)\left(5-4i\right)}{41}$$

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

$$\frac{4\times 5+4\times \left(-4i\right)+4i\times 5+4\left(-4\right)i^{2}}{41}$$

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

$$\frac{4\times 5+4\times \left(-4i\right)+4i\times 5+4\left(-4\right)\left(-1\right)}{41}$$

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

$$\frac{20-16i+20i+16}{41}$$

Combine the real and imaginary parts in 20-16i+20i+16.

$$\frac{20+16+\left(-16+20\right)i}{41}$$

Do the additions in 20+16+\left(-16+20\right)i.

$$\frac{36+4i}{41}$$

Divide 36+4i by 41 to get \frac{36}{41}+\frac{4}{41}i.

$$\frac{36}{41}+\frac{4}{41}i$$

Real Part

$$\frac{36}{41} = 0.8780487804878049$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(4+4i\right)\left(5-4i\right)}{41})$$

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

$$Re(\frac{4\times 5+4\times \left(-4i\right)+4i\times 5+4\left(-4\right)i^{2}}{41})$$

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

$$Re(\frac{4\times 5+4\times \left(-4i\right)+4i\times 5+4\left(-4\right)\left(-1\right)}{41})$$

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

$$Re(\frac{20-16i+20i+16}{41})$$

Combine the real and imaginary parts in 20-16i+20i+16.

$$Re(\frac{20+16+\left(-16+20\right)i}{41})$$

Do the additions in 20+16+\left(-16+20\right)i.

$$Re(\frac{36+4i}{41})$$

Divide 36+4i by 41 to get \frac{36}{41}+\frac{4}{41}i.

$$Re(\frac{36}{41}+\frac{4}{41}i)$$

The real part of \frac{36}{41}+\frac{4}{41}i is \frac{36}{41}.

$$\frac{36}{41}$$