Solve left(130+59.15iright)left(30+13.65iright)/130+59.15i+30+13.65i

$$\frac{ \left( 130+59.15i \right) \left( 30+13.65i \right) }{ 130+59.15i+30+13.65i }$$

Evaluate

$$24.375+11.090625i$$

Solution Steps

Multiply complex numbers 130+59.15i and 30+13.65i like you multiply binomials.

$$\frac{130\times 30+130\times \left(13.65i\right)+59.15i\times 30+59.15\times 13.65i^{2}}{130+59.15i+30+13.65i}$$

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

$$\frac{130\times 30+130\times \left(13.65i\right)+59.15i\times 30+59.15\times 13.65\left(-1\right)}{130+59.15i+30+13.65i}$$

Do the multiplications in 130\times 30+130\times \left(13.65i\right)+59.15i\times 30+59.15\times 13.65\left(-1\right).

$$\frac{3900+1774.5i+1774.5i-807.3975}{130+59.15i+30+13.65i}$$

Combine the real and imaginary parts in 3900+1774.5i+1774.5i-807.3975.

$$\frac{3900-807.3975+\left(1774.5+1774.5\right)i}{130+59.15i+30+13.65i}$$

Do the additions in 3900-807.3975+\left(1774.5+1774.5\right)i.

$$\frac{3092.6025+3549i}{130+59.15i+30+13.65i}$$

Combine the real and imaginary parts in 130+59.15i+30+13.65i.

$$\frac{3092.6025+3549i}{130+30+\left(59.15+13.65\right)i}$$

Do the additions in 130+30+\left(59.15+13.65\right)i.

$$\frac{3092.6025+3549i}{160+72.8i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 160-72.8i.

$$\frac{\left(3092.6025+3549i\right)\left(160-72.8i\right)}{\left(160+72.8i\right)\left(160-72.8i\right)}$$

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

$$\frac{\left(3092.6025+3549i\right)\left(160-72.8i\right)}{160^{2}-72.8^{2}i^{2}}$$

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

$$\frac{\left(3092.6025+3549i\right)\left(160-72.8i\right)}{30899.84}$$

Multiply complex numbers 3092.6025+3549i and 160-72.8i like you multiply binomials.

$$\frac{3092.6025\times 160+3092.6025\times \left(-72.8i\right)+3549i\times 160+3549\left(-72.8\right)i^{2}}{30899.84}$$

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

$$\frac{3092.6025\times 160+3092.6025\times \left(-72.8i\right)+3549i\times 160+3549\left(-72.8\right)\left(-1\right)}{30899.84}$$

Do the multiplications in 3092.6025\times 160+3092.6025\times \left(-72.8i\right)+3549i\times 160+3549\left(-72.8\right)\left(-1\right).

$$\frac{494816.4-225141.462i+567840i+258367.2}{30899.84}$$

Combine the real and imaginary parts in 494816.4-225141.462i+567840i+258367.2.

$$\frac{494816.4+258367.2+\left(-225141.462+567840\right)i}{30899.84}$$

Do the additions in 494816.4+258367.2+\left(-225141.462+567840\right)i.

$$\frac{753183.6+342698.538i}{30899.84}$$

Divide 753183.6+342698.538i by 30899.84 to get 24.375+11.090625i.

$$24.375+11.090625i$$

Real Part

$$24.375$$

Solution Steps

Multiply complex numbers 130+59.15i and 30+13.65i like you multiply binomials.

$$Re(\frac{130\times 30+130\times \left(13.65i\right)+59.15i\times 30+59.15\times 13.65i^{2}}{130+59.15i+30+13.65i})$$

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

$$Re(\frac{130\times 30+130\times \left(13.65i\right)+59.15i\times 30+59.15\times 13.65\left(-1\right)}{130+59.15i+30+13.65i})$$

Do the multiplications in 130\times 30+130\times \left(13.65i\right)+59.15i\times 30+59.15\times 13.65\left(-1\right).

$$Re(\frac{3900+1774.5i+1774.5i-807.3975}{130+59.15i+30+13.65i})$$

Combine the real and imaginary parts in 3900+1774.5i+1774.5i-807.3975.

$$Re(\frac{3900-807.3975+\left(1774.5+1774.5\right)i}{130+59.15i+30+13.65i})$$

Do the additions in 3900-807.3975+\left(1774.5+1774.5\right)i.

$$Re(\frac{3092.6025+3549i}{130+59.15i+30+13.65i})$$

Combine the real and imaginary parts in 130+59.15i+30+13.65i.

$$Re(\frac{3092.6025+3549i}{130+30+\left(59.15+13.65\right)i})$$

Do the additions in 130+30+\left(59.15+13.65\right)i.

$$Re(\frac{3092.6025+3549i}{160+72.8i})$$

Multiply both numerator and denominator of \frac{3092.6025+3549i}{160+72.8i} by the complex conjugate of the denominator, 160-72.8i.

$$Re(\frac{\left(3092.6025+3549i\right)\left(160-72.8i\right)}{\left(160+72.8i\right)\left(160-72.8i\right)})$$

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

$$Re(\frac{\left(3092.6025+3549i\right)\left(160-72.8i\right)}{160^{2}-72.8^{2}i^{2}})$$

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

$$Re(\frac{\left(3092.6025+3549i\right)\left(160-72.8i\right)}{30899.84})$$

Multiply complex numbers 3092.6025+3549i and 160-72.8i like you multiply binomials.

$$Re(\frac{3092.6025\times 160+3092.6025\times \left(-72.8i\right)+3549i\times 160+3549\left(-72.8\right)i^{2}}{30899.84})$$

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

$$Re(\frac{3092.6025\times 160+3092.6025\times \left(-72.8i\right)+3549i\times 160+3549\left(-72.8\right)\left(-1\right)}{30899.84})$$

Do the multiplications in 3092.6025\times 160+3092.6025\times \left(-72.8i\right)+3549i\times 160+3549\left(-72.8\right)\left(-1\right).

$$Re(\frac{494816.4-225141.462i+567840i+258367.2}{30899.84})$$

Combine the real and imaginary parts in 494816.4-225141.462i+567840i+258367.2.

$$Re(\frac{494816.4+258367.2+\left(-225141.462+567840\right)i}{30899.84})$$

Do the additions in 494816.4+258367.2+\left(-225141.462+567840\right)i.

$$Re(\frac{753183.6+342698.538i}{30899.84})$$

Divide 753183.6+342698.538i by 30899.84 to get 24.375+11.090625i.

$$Re(24.375+11.090625i)$$

The real part of 24.375+11.090625i is 24.375.

$$24.375$$