Solve left(130+5915iright)left(30+1365iright)/130+5915i+30+1365i

$$\frac{ \left( 130+5915i \right) \left( 30+1365i \right) }{ 130+5915i+30+1365i }$$

Evaluate

$$\frac{195}{8}+\frac{17745}{16}i=24.375+1109.0625i$$

Solution Steps

Multiply complex numbers 130+5915i and 30+1365i like you multiply binomials.

$$\frac{130\times 30+130\times \left(1365i\right)+5915i\times 30+5915\times 1365i^{2}}{130+5915i+30+1365i}$$

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

$$\frac{130\times 30+130\times \left(1365i\right)+5915i\times 30+5915\times 1365\left(-1\right)}{130+5915i+30+1365i}$$

Do the multiplications in 130\times 30+130\times \left(1365i\right)+5915i\times 30+5915\times 1365\left(-1\right).

$$\frac{3900+177450i+177450i-8073975}{130+5915i+30+1365i}$$

Combine the real and imaginary parts in 3900+177450i+177450i-8073975.

$$\frac{3900-8073975+\left(177450+177450\right)i}{130+5915i+30+1365i}$$

Do the additions in 3900-8073975+\left(177450+177450\right)i.

$$\frac{-8070075+354900i}{130+5915i+30+1365i}$$

Combine the real and imaginary parts in 130+5915i+30+1365i.

$$\frac{-8070075+354900i}{130+30+\left(5915+1365\right)i}$$

Do the additions in 130+30+\left(5915+1365\right)i.

$$\frac{-8070075+354900i}{160+7280i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 160-7280i.

$$\frac{\left(-8070075+354900i\right)\left(160-7280i\right)}{\left(160+7280i\right)\left(160-7280i\right)}$$

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

$$\frac{\left(-8070075+354900i\right)\left(160-7280i\right)}{160^{2}-7280^{2}i^{2}}$$

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

$$\frac{\left(-8070075+354900i\right)\left(160-7280i\right)}{53024000}$$

Multiply complex numbers -8070075+354900i and 160-7280i like you multiply binomials.

$$\frac{-8070075\times 160-8070075\times \left(-7280i\right)+354900i\times 160+354900\left(-7280\right)i^{2}}{53024000}$$

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

$$\frac{-8070075\times 160-8070075\times \left(-7280i\right)+354900i\times 160+354900\left(-7280\right)\left(-1\right)}{53024000}$$

Do the multiplications in -8070075\times 160-8070075\times \left(-7280i\right)+354900i\times 160+354900\left(-7280\right)\left(-1\right).

$$\frac{-1291212000+58750146000i+56784000i+2583672000}{53024000}$$

Combine the real and imaginary parts in -1291212000+58750146000i+56784000i+2583672000.

$$\frac{-1291212000+2583672000+\left(58750146000+56784000\right)i}{53024000}$$

Do the additions in -1291212000+2583672000+\left(58750146000+56784000\right)i.

$$\frac{1292460000+58806930000i}{53024000}$$

Divide 1292460000+58806930000i by 53024000 to get \frac{195}{8}+\frac{17745}{16}i.

$$\frac{195}{8}+\frac{17745}{16}i$$

Real Part

$$\frac{195}{8} = 24\frac{3}{8} = 24.375$$

Solution Steps

Multiply complex numbers 130+5915i and 30+1365i like you multiply binomials.

$$Re(\frac{130\times 30+130\times \left(1365i\right)+5915i\times 30+5915\times 1365i^{2}}{130+5915i+30+1365i})$$

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

$$Re(\frac{130\times 30+130\times \left(1365i\right)+5915i\times 30+5915\times 1365\left(-1\right)}{130+5915i+30+1365i})$$

Do the multiplications in 130\times 30+130\times \left(1365i\right)+5915i\times 30+5915\times 1365\left(-1\right).

$$Re(\frac{3900+177450i+177450i-8073975}{130+5915i+30+1365i})$$

Combine the real and imaginary parts in 3900+177450i+177450i-8073975.

$$Re(\frac{3900-8073975+\left(177450+177450\right)i}{130+5915i+30+1365i})$$

Do the additions in 3900-8073975+\left(177450+177450\right)i.

$$Re(\frac{-8070075+354900i}{130+5915i+30+1365i})$$

Combine the real and imaginary parts in 130+5915i+30+1365i.

$$Re(\frac{-8070075+354900i}{130+30+\left(5915+1365\right)i})$$

Do the additions in 130+30+\left(5915+1365\right)i.

$$Re(\frac{-8070075+354900i}{160+7280i})$$

Multiply both numerator and denominator of \frac{-8070075+354900i}{160+7280i} by the complex conjugate of the denominator, 160-7280i.

$$Re(\frac{\left(-8070075+354900i\right)\left(160-7280i\right)}{\left(160+7280i\right)\left(160-7280i\right)})$$

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

$$Re(\frac{\left(-8070075+354900i\right)\left(160-7280i\right)}{160^{2}-7280^{2}i^{2}})$$

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

$$Re(\frac{\left(-8070075+354900i\right)\left(160-7280i\right)}{53024000})$$

Multiply complex numbers -8070075+354900i and 160-7280i like you multiply binomials.

$$Re(\frac{-8070075\times 160-8070075\times \left(-7280i\right)+354900i\times 160+354900\left(-7280\right)i^{2}}{53024000})$$

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

$$Re(\frac{-8070075\times 160-8070075\times \left(-7280i\right)+354900i\times 160+354900\left(-7280\right)\left(-1\right)}{53024000})$$

Do the multiplications in -8070075\times 160-8070075\times \left(-7280i\right)+354900i\times 160+354900\left(-7280\right)\left(-1\right).

$$Re(\frac{-1291212000+58750146000i+56784000i+2583672000}{53024000})$$

Combine the real and imaginary parts in -1291212000+58750146000i+56784000i+2583672000.

$$Re(\frac{-1291212000+2583672000+\left(58750146000+56784000\right)i}{53024000})$$

Do the additions in -1291212000+2583672000+\left(58750146000+56784000\right)i.

$$Re(\frac{1292460000+58806930000i}{53024000})$$

Divide 1292460000+58806930000i by 53024000 to get \frac{195}{8}+\frac{17745}{16}i.

$$Re(\frac{195}{8}+\frac{17745}{16}i)$$

The real part of \frac{195}{8}+\frac{17745}{16}i is \frac{195}{8}.

$$\frac{195}{8}$$