Solve frac{(329+190texttt{i})*(3+4texttt{i})}{329+3+190texttt{i}+4texttt{i}}

$$\frac{ (329+190 \texttt{i} ) \times (3+4 \texttt{i} ) }{ 329+3+190 \texttt{i} +4 \texttt{i} }$$

Evaluate

$$\frac{110312}{36965}+\frac{291057}{73930}i\approx 2.984228324+3.936926823i$$

Solution Steps

Multiply complex numbers 329+190i and 3+4i like you multiply binomials.

$$\frac{329\times 3+329\times \left(4i\right)+190i\times 3+190\times 4i^{2}}{329+3+190i+4i}$$

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

$$\frac{329\times 3+329\times \left(4i\right)+190i\times 3+190\times 4\left(-1\right)}{329+3+190i+4i}$$

Do the multiplications in 329\times 3+329\times \left(4i\right)+190i\times 3+190\times 4\left(-1\right).

$$\frac{987+1316i+570i-760}{329+3+190i+4i}$$

Combine the real and imaginary parts in 987+1316i+570i-760.

$$\frac{987-760+\left(1316+570\right)i}{329+3+190i+4i}$$

Do the additions in 987-760+\left(1316+570\right)i.

$$\frac{227+1886i}{329+3+190i+4i}$$

Combine the real and imaginary parts in 329+3+190i+4i.

$$\frac{227+1886i}{329+3+\left(190+4\right)i}$$

Do the additions in 329+3+\left(190+4\right)i.

$$\frac{227+1886i}{332+194i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 332-194i.

$$\frac{\left(227+1886i\right)\left(332-194i\right)}{\left(332+194i\right)\left(332-194i\right)}$$

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

$$\frac{\left(227+1886i\right)\left(332-194i\right)}{332^{2}-194^{2}i^{2}}$$

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

$$\frac{\left(227+1886i\right)\left(332-194i\right)}{147860}$$

Multiply complex numbers 227+1886i and 332-194i like you multiply binomials.

$$\frac{227\times 332+227\times \left(-194i\right)+1886i\times 332+1886\left(-194\right)i^{2}}{147860}$$

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

$$\frac{227\times 332+227\times \left(-194i\right)+1886i\times 332+1886\left(-194\right)\left(-1\right)}{147860}$$

Do the multiplications in 227\times 332+227\times \left(-194i\right)+1886i\times 332+1886\left(-194\right)\left(-1\right).

$$\frac{75364-44038i+626152i+365884}{147860}$$

Combine the real and imaginary parts in 75364-44038i+626152i+365884.

$$\frac{75364+365884+\left(-44038+626152\right)i}{147860}$$

Do the additions in 75364+365884+\left(-44038+626152\right)i.

$$\frac{441248+582114i}{147860}$$

Divide 441248+582114i by 147860 to get \frac{110312}{36965}+\frac{291057}{73930}i.

$$\frac{110312}{36965}+\frac{291057}{73930}i$$

Real Part

$$\frac{110312}{36965} = 2\frac{36382}{36965} = 2.9842283240903558$$

Solution Steps

Multiply complex numbers 329+190i and 3+4i like you multiply binomials.

$$Re(\frac{329\times 3+329\times \left(4i\right)+190i\times 3+190\times 4i^{2}}{329+3+190i+4i})$$

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

$$Re(\frac{329\times 3+329\times \left(4i\right)+190i\times 3+190\times 4\left(-1\right)}{329+3+190i+4i})$$

Do the multiplications in 329\times 3+329\times \left(4i\right)+190i\times 3+190\times 4\left(-1\right).

$$Re(\frac{987+1316i+570i-760}{329+3+190i+4i})$$

Combine the real and imaginary parts in 987+1316i+570i-760.

$$Re(\frac{987-760+\left(1316+570\right)i}{329+3+190i+4i})$$

Do the additions in 987-760+\left(1316+570\right)i.

$$Re(\frac{227+1886i}{329+3+190i+4i})$$

Combine the real and imaginary parts in 329+3+190i+4i.

$$Re(\frac{227+1886i}{329+3+\left(190+4\right)i})$$

Do the additions in 329+3+\left(190+4\right)i.

$$Re(\frac{227+1886i}{332+194i})$$

Multiply both numerator and denominator of \frac{227+1886i}{332+194i} by the complex conjugate of the denominator, 332-194i.

$$Re(\frac{\left(227+1886i\right)\left(332-194i\right)}{\left(332+194i\right)\left(332-194i\right)})$$

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

$$Re(\frac{\left(227+1886i\right)\left(332-194i\right)}{332^{2}-194^{2}i^{2}})$$

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

$$Re(\frac{\left(227+1886i\right)\left(332-194i\right)}{147860})$$

Multiply complex numbers 227+1886i and 332-194i like you multiply binomials.

$$Re(\frac{227\times 332+227\times \left(-194i\right)+1886i\times 332+1886\left(-194\right)i^{2}}{147860})$$

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

$$Re(\frac{227\times 332+227\times \left(-194i\right)+1886i\times 332+1886\left(-194\right)\left(-1\right)}{147860})$$

Do the multiplications in 227\times 332+227\times \left(-194i\right)+1886i\times 332+1886\left(-194\right)\left(-1\right).

$$Re(\frac{75364-44038i+626152i+365884}{147860})$$

Combine the real and imaginary parts in 75364-44038i+626152i+365884.

$$Re(\frac{75364+365884+\left(-44038+626152\right)i}{147860})$$

Do the additions in 75364+365884+\left(-44038+626152\right)i.

$$Re(\frac{441248+582114i}{147860})$$

Divide 441248+582114i by 147860 to get \frac{110312}{36965}+\frac{291057}{73930}i.

$$Re(\frac{110312}{36965}+\frac{291057}{73930}i)$$

The real part of \frac{110312}{36965}+\frac{291057}{73930}i is \frac{110312}{36965}.

$$\frac{110312}{36965}$$