Solve 10-10i/6-2i

$$\frac { 10 - 10 i } { 6 - 2 i }$$

Evaluate

$$2-i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 6+2i.

$$\frac{\left(10-10i\right)\left(6+2i\right)}{\left(6-2i\right)\left(6+2i\right)}$$

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

$$\frac{\left(10-10i\right)\left(6+2i\right)}{6^{2}-2^{2}i^{2}}$$

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

$$\frac{\left(10-10i\right)\left(6+2i\right)}{40}$$

Multiply complex numbers 10-10i and 6+2i like you multiply binomials.

$$\frac{10\times 6+10\times \left(2i\right)-10i\times 6-10\times 2i^{2}}{40}$$

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

$$\frac{10\times 6+10\times \left(2i\right)-10i\times 6-10\times 2\left(-1\right)}{40}$$

Do the multiplications in 10\times 6+10\times \left(2i\right)-10i\times 6-10\times 2\left(-1\right).

$$\frac{60+20i-60i+20}{40}$$

Combine the real and imaginary parts in 60+20i-60i+20.

$$\frac{60+20+\left(20-60\right)i}{40}$$

Do the additions in 60+20+\left(20-60\right)i.

$$\frac{80-40i}{40}$$

Divide 80-40i by 40 to get 2-i.

$$2-i$$

Real Part

$$2$$

Solution Steps

Multiply both numerator and denominator of \frac{10-10i}{6-2i} by the complex conjugate of the denominator, 6+2i.

$$Re(\frac{\left(10-10i\right)\left(6+2i\right)}{\left(6-2i\right)\left(6+2i\right)})$$

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

$$Re(\frac{\left(10-10i\right)\left(6+2i\right)}{6^{2}-2^{2}i^{2}})$$

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

$$Re(\frac{\left(10-10i\right)\left(6+2i\right)}{40})$$

Multiply complex numbers 10-10i and 6+2i like you multiply binomials.

$$Re(\frac{10\times 6+10\times \left(2i\right)-10i\times 6-10\times 2i^{2}}{40})$$

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

$$Re(\frac{10\times 6+10\times \left(2i\right)-10i\times 6-10\times 2\left(-1\right)}{40})$$

Do the multiplications in 10\times 6+10\times \left(2i\right)-10i\times 6-10\times 2\left(-1\right).

$$Re(\frac{60+20i-60i+20}{40})$$

Combine the real and imaginary parts in 60+20i-60i+20.

$$Re(\frac{60+20+\left(20-60\right)i}{40})$$

Do the additions in 60+20+\left(20-60\right)i.

$$Re(\frac{80-40i}{40})$$

Divide 80-40i by 40 to get 2-i.

$$Re(2-i)$$

The real part of 2-i is 2.

$$2$$