Solve 33-i/3-i

$$\frac { 33 - i } { 3 - i }$$

Evaluate

$$10+3i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3+i.

$$\frac{\left(33-i\right)\left(3+i\right)}{\left(3-i\right)\left(3+i\right)}$$

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

$$\frac{\left(33-i\right)\left(3+i\right)}{3^{2}-i^{2}}$$

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

$$\frac{\left(33-i\right)\left(3+i\right)}{10}$$

Multiply complex numbers 33-i and 3+i like you multiply binomials.

$$\frac{33\times 3+33i-i\times 3-i^{2}}{10}$$

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

$$\frac{33\times 3+33i-i\times 3-\left(-1\right)}{10}$$

Do the multiplications in 33\times 3+33i-i\times 3-\left(-1\right).

$$\frac{99+33i-3i+1}{10}$$

Combine the real and imaginary parts in 99+33i-3i+1.

$$\frac{99+1+\left(33-3\right)i}{10}$$

Do the additions in 99+1+\left(33-3\right)i.

$$\frac{100+30i}{10}$$

Divide 100+30i by 10 to get 10+3i.

$$10+3i$$

Real Part

$$10$$

Solution Steps

Multiply both numerator and denominator of \frac{33-i}{3-i} by the complex conjugate of the denominator, 3+i.

$$Re(\frac{\left(33-i\right)\left(3+i\right)}{\left(3-i\right)\left(3+i\right)})$$

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

$$Re(\frac{\left(33-i\right)\left(3+i\right)}{3^{2}-i^{2}})$$

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

$$Re(\frac{\left(33-i\right)\left(3+i\right)}{10})$$

Multiply complex numbers 33-i and 3+i like you multiply binomials.

$$Re(\frac{33\times 3+33i-i\times 3-i^{2}}{10})$$

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

$$Re(\frac{33\times 3+33i-i\times 3-\left(-1\right)}{10})$$

Do the multiplications in 33\times 3+33i-i\times 3-\left(-1\right).

$$Re(\frac{99+33i-3i+1}{10})$$

Combine the real and imaginary parts in 99+33i-3i+1.

$$Re(\frac{99+1+\left(33-3\right)i}{10})$$

Do the additions in 99+1+\left(33-3\right)i.

$$Re(\frac{100+30i}{10})$$

Divide 100+30i by 10 to get 10+3i.

$$Re(10+3i)$$

The real part of 10+3i is 10.

$$10$$