Solve -9-(-1/-8-i-1

$$\frac { - 9 - ( - 1 } { - 8 - i - 1 }$$

Evaluate

$$-\frac{747}{82}+\frac{1}{82}i\approx -9.109756098+0.012195122i$$

Solution Steps

Subtract 1 from -8-i by subtracting corresponding real and imaginary parts.

$$-9-\frac{-1}{-8-1-i}$$

Subtract 1 from -8 to get -9.

$$-9-\frac{-1}{-9-i}$$

Multiply both numerator and denominator of \frac{-1}{-9-i} by the complex conjugate of the denominator, -9+i.

$$-9-\frac{-\left(-9+i\right)}{\left(-9-i\right)\left(-9+i\right)}$$

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

$$-9-\frac{-\left(-9+i\right)}{\left(-9\right)^{2}-i^{2}}$$

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

$$-9-\frac{-\left(-9+i\right)}{82}$$

Multiply -1 and -9+i to get 9-i.

$$-9-\frac{9-i}{82}$$

Divide 9-i by 82 to get \frac{9}{82}-\frac{1}{82}i.

$$-9+\left(-\frac{9}{82}+\frac{1}{82}i\right)$$

Combine the real and imaginary parts in numbers -9 and -\frac{9}{82}+\frac{1}{82}i.

$$-9-\frac{9}{82}+\frac{1}{82}i$$

Add -9 to -\frac{9}{82}.

$$-\frac{747}{82}+\frac{1}{82}i$$

Real Part

$$-\frac{747}{82} = -9\frac{9}{82} = -9.109756097560975$$

Solution Steps

Subtract 1 from -8-i by subtracting corresponding real and imaginary parts.

$$Re(-9-\frac{-1}{-8-1-i})$$

Subtract 1 from -8 to get -9.

$$Re(-9-\frac{-1}{-9-i})$$

Multiply both numerator and denominator of \frac{-1}{-9-i} by the complex conjugate of the denominator, -9+i.

$$Re(-9-\frac{-\left(-9+i\right)}{\left(-9-i\right)\left(-9+i\right)})$$

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

$$Re(-9-\frac{-\left(-9+i\right)}{\left(-9\right)^{2}-i^{2}})$$

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

$$Re(-9-\frac{-\left(-9+i\right)}{82})$$

Multiply -1 and -9+i to get 9-i.

$$Re(-9-\frac{9-i}{82})$$

Divide 9-i by 82 to get \frac{9}{82}-\frac{1}{82}i.

$$Re(-9+\left(-\frac{9}{82}+\frac{1}{82}i\right))$$

Combine the real and imaginary parts in numbers -9 and -\frac{9}{82}+\frac{1}{82}i.

$$Re(-9-\frac{9}{82}+\frac{1}{82}i)$$

Add -9 to -\frac{9}{82}.

$$Re(-\frac{747}{82}+\frac{1}{82}i)$$

The real part of -\frac{747}{82}+\frac{1}{82}i is -\frac{747}{82}.

$$-\frac{747}{82}$$