Solve -7-4i/-8+i

$$\frac { - 7 - 4 i } { - 8 + i }$$

Evaluate

$$\frac{4}{5}+\frac{3}{5}i=0.8+0.6i$$

Solution Steps

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

$$\frac{\left(-7-4i\right)\left(-8-i\right)}{\left(-8+i\right)\left(-8-i\right)}$$

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

$$\frac{\left(-7-4i\right)\left(-8-i\right)}{\left(-8\right)^{2}-i^{2}}$$

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

$$\frac{\left(-7-4i\right)\left(-8-i\right)}{65}$$

Multiply complex numbers -7-4i and -8-i like you multiply binomials.

$$\frac{-7\left(-8\right)-7\left(-i\right)-4i\left(-8\right)-4\left(-1\right)i^{2}}{65}$$

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

$$\frac{-7\left(-8\right)-7\left(-i\right)-4i\left(-8\right)-4\left(-1\right)\left(-1\right)}{65}$$

Do the multiplications in -7\left(-8\right)-7\left(-i\right)-4i\left(-8\right)-4\left(-1\right)\left(-1\right).

$$\frac{56+7i+32i-4}{65}$$

Combine the real and imaginary parts in 56+7i+32i-4.

$$\frac{56-4+\left(7+32\right)i}{65}$$

Do the additions in 56-4+\left(7+32\right)i.

$$\frac{52+39i}{65}$$

Divide 52+39i by 65 to get \frac{4}{5}+\frac{3}{5}i.

$$\frac{4}{5}+\frac{3}{5}i$$

Real Part

$$\frac{4}{5} = 0.8$$

Solution Steps

Multiply both numerator and denominator of \frac{-7-4i}{-8+i} by the complex conjugate of the denominator, -8-i.

$$Re(\frac{\left(-7-4i\right)\left(-8-i\right)}{\left(-8+i\right)\left(-8-i\right)})$$

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

$$Re(\frac{\left(-7-4i\right)\left(-8-i\right)}{\left(-8\right)^{2}-i^{2}})$$

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

$$Re(\frac{\left(-7-4i\right)\left(-8-i\right)}{65})$$

Multiply complex numbers -7-4i and -8-i like you multiply binomials.

$$Re(\frac{-7\left(-8\right)-7\left(-i\right)-4i\left(-8\right)-4\left(-1\right)i^{2}}{65})$$

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

$$Re(\frac{-7\left(-8\right)-7\left(-i\right)-4i\left(-8\right)-4\left(-1\right)\left(-1\right)}{65})$$

Do the multiplications in -7\left(-8\right)-7\left(-i\right)-4i\left(-8\right)-4\left(-1\right)\left(-1\right).

$$Re(\frac{56+7i+32i-4}{65})$$

Combine the real and imaginary parts in 56+7i+32i-4.

$$Re(\frac{56-4+\left(7+32\right)i}{65})$$

Do the additions in 56-4+\left(7+32\right)i.

$$Re(\frac{52+39i}{65})$$

Divide 52+39i by 65 to get \frac{4}{5}+\frac{3}{5}i.

$$Re(\frac{4}{5}+\frac{3}{5}i)$$

The real part of \frac{4}{5}+\frac{3}{5}i is \frac{4}{5}.

$$\frac{4}{5}$$