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

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

Calcular

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

Pasos da solución

Multiplica os números complexos 130+5915i e 30+1365i igual que se multiplican os binomios.

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

Por definición, i^{2} é -1.

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

Fai as multiplicacións en 130\times 30+130\times \left(1365i\right)+5915i\times 30+5915\times 1365\left(-1\right).

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

Combina as partes reais e imaxinarias en 3900+177450i+177450i-8073975.

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

Fai as sumas en 3900-8073975+\left(177450+177450\right)i.

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

Combina as partes reais e imaxinarias en 130+5915i+30+1365i.

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

Fai as sumas en 130+30+\left(5915+1365\right)i.

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

Multiplica o numerador e o denominador polo conxugado complexo do denominador 160-7280i.

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

A multiplicación pódese transformar na diferencia de cadrados mediante a regra: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Por definición, i^{2} é -1. Calcula o denominador.

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

Multiplica os números complexos -8070075+354900i e 160-7280i igual que se multiplican os binomios.

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

Por definición, i^{2} é -1.

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

Fai as multiplicacións en -8070075\times 160-8070075\times \left(-7280i\right)+354900i\times 160+354900\left(-7280\right)\left(-1\right).

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

Combina as partes reais e imaxinarias en -1291212000+58750146000i+56784000i+2583672000.

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

Fai as sumas en -1291212000+2583672000+\left(58750146000+56784000\right)i.

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

Divide 1292460000+58806930000i entre 53024000 para obter \frac{195}{8}+\frac{17745}{16}i.

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

Parte real

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

Pasos da solución

Multiplica os números complexos 130+5915i e 30+1365i igual que se multiplican os binomios.

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

Por definición, i^{2} é -1.

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

Fai as multiplicacións en 130\times 30+130\times \left(1365i\right)+5915i\times 30+5915\times 1365\left(-1\right).

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

Combina as partes reais e imaxinarias en 3900+177450i+177450i-8073975.

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

Fai as sumas en 3900-8073975+\left(177450+177450\right)i.

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

Combina as partes reais e imaxinarias en 130+5915i+30+1365i.

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

Fai as sumas en 130+30+\left(5915+1365\right)i.

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

Multiplica o numerador e o denominador de \frac{-8070075+354900i}{160+7280i} polo conxugado complexo do denominador, 160-7280i.

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

A multiplicación pódese transformar na diferencia de cadrados mediante a regra: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Por definición, i^{2} é -1. Calcula o denominador.

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

Multiplica os números complexos -8070075+354900i e 160-7280i igual que se multiplican os binomios.

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

Por definición, i^{2} é -1.

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

Fai as multiplicacións en -8070075\times 160-8070075\times \left(-7280i\right)+354900i\times 160+354900\left(-7280\right)\left(-1\right).

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

Combina as partes reais e imaxinarias en -1291212000+58750146000i+56784000i+2583672000.

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

Fai as sumas en -1291212000+2583672000+\left(58750146000+56784000\right)i.

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

Divide 1292460000+58806930000i entre 53024000 para obter \frac{195}{8}+\frac{17745}{16}i.

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

A parte real de \frac{195}{8}+\frac{17745}{16}i é \frac{195}{8}.

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