Solve 9-19i/1-5i

$$\frac { 9 - 19 i } { 1 - 5 i }$$

Evaluate

$$4+i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 1+5i.

$$\frac{\left(9-19i\right)\left(1+5i\right)}{\left(1-5i\right)\left(1+5i\right)}$$

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

$$\frac{\left(9-19i\right)\left(1+5i\right)}{1^{2}-5^{2}i^{2}}$$

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

$$\frac{\left(9-19i\right)\left(1+5i\right)}{26}$$

Multiply complex numbers 9-19i and 1+5i like you multiply binomials.

$$\frac{9\times 1+9\times \left(5i\right)-19i-19\times 5i^{2}}{26}$$

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

$$\frac{9\times 1+9\times \left(5i\right)-19i-19\times 5\left(-1\right)}{26}$$

Do the multiplications in 9\times 1+9\times \left(5i\right)-19i-19\times 5\left(-1\right).

$$\frac{9+45i-19i+95}{26}$$

Combine the real and imaginary parts in 9+45i-19i+95.

$$\frac{9+95+\left(45-19\right)i}{26}$$

Do the additions in 9+95+\left(45-19\right)i.

$$\frac{104+26i}{26}$$

Divide 104+26i by 26 to get 4+i.

$$4+i$$

Real Part

$$4$$

Solution Steps

Multiply both numerator and denominator of \frac{9-19i}{1-5i} by the complex conjugate of the denominator, 1+5i.

$$Re(\frac{\left(9-19i\right)\left(1+5i\right)}{\left(1-5i\right)\left(1+5i\right)})$$

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

$$Re(\frac{\left(9-19i\right)\left(1+5i\right)}{1^{2}-5^{2}i^{2}})$$

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

$$Re(\frac{\left(9-19i\right)\left(1+5i\right)}{26})$$

Multiply complex numbers 9-19i and 1+5i like you multiply binomials.

$$Re(\frac{9\times 1+9\times \left(5i\right)-19i-19\times 5i^{2}}{26})$$

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

$$Re(\frac{9\times 1+9\times \left(5i\right)-19i-19\times 5\left(-1\right)}{26})$$

Do the multiplications in 9\times 1+9\times \left(5i\right)-19i-19\times 5\left(-1\right).

$$Re(\frac{9+45i-19i+95}{26})$$

Combine the real and imaginary parts in 9+45i-19i+95.

$$Re(\frac{9+95+\left(45-19\right)i}{26})$$

Do the additions in 9+95+\left(45-19\right)i.

$$Re(\frac{104+26i}{26})$$

Divide 104+26i by 26 to get 4+i.

$$Re(4+i)$$

The real part of 4+i is 4.

$$4$$