Solve 7-2i/1-3i

$$\frac { 7 - 2 i } { 1 - 3 i }$$

Evaluate

$$\frac{13}{10}+\frac{19}{10}i=1.3+1.9i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 1+3i.

$$\frac{\left(7-2i\right)\left(1+3i\right)}{\left(1-3i\right)\left(1+3i\right)}$$

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

$$\frac{\left(7-2i\right)\left(1+3i\right)}{1^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(7-2i\right)\left(1+3i\right)}{10}$$

Multiply complex numbers 7-2i and 1+3i like you multiply binomials.

$$\frac{7\times 1+7\times \left(3i\right)-2i-2\times 3i^{2}}{10}$$

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

$$\frac{7\times 1+7\times \left(3i\right)-2i-2\times 3\left(-1\right)}{10}$$

Do the multiplications in 7\times 1+7\times \left(3i\right)-2i-2\times 3\left(-1\right).

$$\frac{7+21i-2i+6}{10}$$

Combine the real and imaginary parts in 7+21i-2i+6.

$$\frac{7+6+\left(21-2\right)i}{10}$$

Do the additions in 7+6+\left(21-2\right)i.

$$\frac{13+19i}{10}$$

Divide 13+19i by 10 to get \frac{13}{10}+\frac{19}{10}i.

$$\frac{13}{10}+\frac{19}{10}i$$

Real Part

$$\frac{13}{10} = 1\frac{3}{10} = 1.3$$

Solution Steps

Multiply both numerator and denominator of \frac{7-2i}{1-3i} by the complex conjugate of the denominator, 1+3i.

$$Re(\frac{\left(7-2i\right)\left(1+3i\right)}{\left(1-3i\right)\left(1+3i\right)})$$

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

$$Re(\frac{\left(7-2i\right)\left(1+3i\right)}{1^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(7-2i\right)\left(1+3i\right)}{10})$$

Multiply complex numbers 7-2i and 1+3i like you multiply binomials.

$$Re(\frac{7\times 1+7\times \left(3i\right)-2i-2\times 3i^{2}}{10})$$

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

$$Re(\frac{7\times 1+7\times \left(3i\right)-2i-2\times 3\left(-1\right)}{10})$$

Do the multiplications in 7\times 1+7\times \left(3i\right)-2i-2\times 3\left(-1\right).

$$Re(\frac{7+21i-2i+6}{10})$$

Combine the real and imaginary parts in 7+21i-2i+6.

$$Re(\frac{7+6+\left(21-2\right)i}{10})$$

Do the additions in 7+6+\left(21-2\right)i.

$$Re(\frac{13+19i}{10})$$

Divide 13+19i by 10 to get \frac{13}{10}+\frac{19}{10}i.

$$Re(\frac{13}{10}+\frac{19}{10}i)$$

The real part of \frac{13}{10}+\frac{19}{10}i is \frac{13}{10}.

$$\frac{13}{10}$$