Solve frac{left(5+10texttt{i}right)20}{5+10texttt{i}+20}

$$\frac{ \left( 5+10 \texttt{i} \right) 20 }{ 5+10 \texttt{i} +20 }$$

Evaluate

$$\frac{180}{29}+\frac{160}{29}i\approx 6.206896552+5.517241379i$$

Solution Steps

Multiply 5+10i times 20.

$$\frac{5\times 20+10i\times 20}{5+10i+20}$$

Do the multiplications in 5\times 20+10i\times 20.

$$\frac{100+200i}{5+10i+20}$$

Combine the real and imaginary parts in numbers 5+10i and 20.

$$\frac{100+200i}{5+20+10i}$$

Add 5 to 20.

$$\frac{100+200i}{25+10i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 25-10i.

$$\frac{\left(100+200i\right)\left(25-10i\right)}{\left(25+10i\right)\left(25-10i\right)}$$

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

$$\frac{\left(100+200i\right)\left(25-10i\right)}{25^{2}-10^{2}i^{2}}$$

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

$$\frac{\left(100+200i\right)\left(25-10i\right)}{725}$$

Multiply complex numbers 100+200i and 25-10i like you multiply binomials.

$$\frac{100\times 25+100\times \left(-10i\right)+200i\times 25+200\left(-10\right)i^{2}}{725}$$

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

$$\frac{100\times 25+100\times \left(-10i\right)+200i\times 25+200\left(-10\right)\left(-1\right)}{725}$$

Do the multiplications in 100\times 25+100\times \left(-10i\right)+200i\times 25+200\left(-10\right)\left(-1\right).

$$\frac{2500-1000i+5000i+2000}{725}$$

Combine the real and imaginary parts in 2500-1000i+5000i+2000.

$$\frac{2500+2000+\left(-1000+5000\right)i}{725}$$

Do the additions in 2500+2000+\left(-1000+5000\right)i.

$$\frac{4500+4000i}{725}$$

Divide 4500+4000i by 725 to get \frac{180}{29}+\frac{160}{29}i.

$$\frac{180}{29}+\frac{160}{29}i$$

Real Part

$$\frac{180}{29} = 6\frac{6}{29} = 6.206896551724138$$

Solution Steps

Multiply 5+10i times 20.

$$Re(\frac{5\times 20+10i\times 20}{5+10i+20})$$

Do the multiplications in 5\times 20+10i\times 20.

$$Re(\frac{100+200i}{5+10i+20})$$

Combine the real and imaginary parts in numbers 5+10i and 20.

$$Re(\frac{100+200i}{5+20+10i})$$

Add 5 to 20.

$$Re(\frac{100+200i}{25+10i})$$

Multiply both numerator and denominator of \frac{100+200i}{25+10i} by the complex conjugate of the denominator, 25-10i.

$$Re(\frac{\left(100+200i\right)\left(25-10i\right)}{\left(25+10i\right)\left(25-10i\right)})$$

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

$$Re(\frac{\left(100+200i\right)\left(25-10i\right)}{25^{2}-10^{2}i^{2}})$$

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

$$Re(\frac{\left(100+200i\right)\left(25-10i\right)}{725})$$

Multiply complex numbers 100+200i and 25-10i like you multiply binomials.

$$Re(\frac{100\times 25+100\times \left(-10i\right)+200i\times 25+200\left(-10\right)i^{2}}{725})$$

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

$$Re(\frac{100\times 25+100\times \left(-10i\right)+200i\times 25+200\left(-10\right)\left(-1\right)}{725})$$

Do the multiplications in 100\times 25+100\times \left(-10i\right)+200i\times 25+200\left(-10\right)\left(-1\right).

$$Re(\frac{2500-1000i+5000i+2000}{725})$$

Combine the real and imaginary parts in 2500-1000i+5000i+2000.

$$Re(\frac{2500+2000+\left(-1000+5000\right)i}{725})$$

Do the additions in 2500+2000+\left(-1000+5000\right)i.

$$Re(\frac{4500+4000i}{725})$$

Divide 4500+4000i by 725 to get \frac{180}{29}+\frac{160}{29}i.

$$Re(\frac{180}{29}+\frac{160}{29}i)$$

The real part of \frac{180}{29}+\frac{160}{29}i is \frac{180}{29}.

$$\frac{180}{29}$$