Solve -66-19i/-8-5i

$$\frac { - 66 - 19 i } { - 8 - 5 i }$$

Evaluate

$$7-2i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, -8+5i.

$$\frac{\left(-66-19i\right)\left(-8+5i\right)}{\left(-8-5i\right)\left(-8+5i\right)}$$

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

$$\frac{\left(-66-19i\right)\left(-8+5i\right)}{\left(-8\right)^{2}-5^{2}i^{2}}$$

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

$$\frac{\left(-66-19i\right)\left(-8+5i\right)}{89}$$

Multiply complex numbers -66-19i and -8+5i like you multiply binomials.

$$\frac{-66\left(-8\right)-66\times \left(5i\right)-19i\left(-8\right)-19\times 5i^{2}}{89}$$

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

$$\frac{-66\left(-8\right)-66\times \left(5i\right)-19i\left(-8\right)-19\times 5\left(-1\right)}{89}$$

Do the multiplications in -66\left(-8\right)-66\times \left(5i\right)-19i\left(-8\right)-19\times 5\left(-1\right).

$$\frac{528-330i+152i+95}{89}$$

Combine the real and imaginary parts in 528-330i+152i+95.

$$\frac{528+95+\left(-330+152\right)i}{89}$$

Do the additions in 528+95+\left(-330+152\right)i.

$$\frac{623-178i}{89}$$

Divide 623-178i by 89 to get 7-2i.

$$7-2i$$

Real Part

$$7$$

Solution Steps

Multiply both numerator and denominator of \frac{-66-19i}{-8-5i} by the complex conjugate of the denominator, -8+5i.

$$Re(\frac{\left(-66-19i\right)\left(-8+5i\right)}{\left(-8-5i\right)\left(-8+5i\right)})$$

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

$$Re(\frac{\left(-66-19i\right)\left(-8+5i\right)}{\left(-8\right)^{2}-5^{2}i^{2}})$$

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

$$Re(\frac{\left(-66-19i\right)\left(-8+5i\right)}{89})$$

Multiply complex numbers -66-19i and -8+5i like you multiply binomials.

$$Re(\frac{-66\left(-8\right)-66\times \left(5i\right)-19i\left(-8\right)-19\times 5i^{2}}{89})$$

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

$$Re(\frac{-66\left(-8\right)-66\times \left(5i\right)-19i\left(-8\right)-19\times 5\left(-1\right)}{89})$$

Do the multiplications in -66\left(-8\right)-66\times \left(5i\right)-19i\left(-8\right)-19\times 5\left(-1\right).

$$Re(\frac{528-330i+152i+95}{89})$$

Combine the real and imaginary parts in 528-330i+152i+95.

$$Re(\frac{528+95+\left(-330+152\right)i}{89})$$

Do the additions in 528+95+\left(-330+152\right)i.

$$Re(\frac{623-178i}{89})$$

Divide 623-178i by 89 to get 7-2i.

$$Re(7-2i)$$

The real part of 7-2i is 7.

$$7$$