Solve (3-5i)div(2-5i)=

$$( 3 - 5 i ) \div ( 2 - 5 i ) =$$

Evaluate

$$\frac{31}{29}+\frac{5}{29}i\approx 1.068965517+0.172413793i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 2+5i.

$$\frac{\left(3-5i\right)\left(2+5i\right)}{\left(2-5i\right)\left(2+5i\right)}$$

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

$$\frac{\left(3-5i\right)\left(2+5i\right)}{2^{2}-5^{2}i^{2}}$$

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

$$\frac{\left(3-5i\right)\left(2+5i\right)}{29}$$

Multiply complex numbers 3-5i and 2+5i like you multiply binomials.

$$\frac{3\times 2+3\times \left(5i\right)-5i\times 2-5\times 5i^{2}}{29}$$

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

$$\frac{3\times 2+3\times \left(5i\right)-5i\times 2-5\times 5\left(-1\right)}{29}$$

Do the multiplications in 3\times 2+3\times \left(5i\right)-5i\times 2-5\times 5\left(-1\right).

$$\frac{6+15i-10i+25}{29}$$

Combine the real and imaginary parts in 6+15i-10i+25.

$$\frac{6+25+\left(15-10\right)i}{29}$$

Do the additions in 6+25+\left(15-10\right)i.

$$\frac{31+5i}{29}$$

Divide 31+5i by 29 to get \frac{31}{29}+\frac{5}{29}i.

$$\frac{31}{29}+\frac{5}{29}i$$

Real Part

$$\frac{31}{29} = 1\frac{2}{29} = 1.0689655172413792$$

Solution Steps

Multiply both numerator and denominator of \frac{3-5i}{2-5i} by the complex conjugate of the denominator, 2+5i.

$$Re(\frac{\left(3-5i\right)\left(2+5i\right)}{\left(2-5i\right)\left(2+5i\right)})$$

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

$$Re(\frac{\left(3-5i\right)\left(2+5i\right)}{2^{2}-5^{2}i^{2}})$$

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

$$Re(\frac{\left(3-5i\right)\left(2+5i\right)}{29})$$

Multiply complex numbers 3-5i and 2+5i like you multiply binomials.

$$Re(\frac{3\times 2+3\times \left(5i\right)-5i\times 2-5\times 5i^{2}}{29})$$

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

$$Re(\frac{3\times 2+3\times \left(5i\right)-5i\times 2-5\times 5\left(-1\right)}{29})$$

Do the multiplications in 3\times 2+3\times \left(5i\right)-5i\times 2-5\times 5\left(-1\right).

$$Re(\frac{6+15i-10i+25}{29})$$

Combine the real and imaginary parts in 6+15i-10i+25.

$$Re(\frac{6+25+\left(15-10\right)i}{29})$$

Do the additions in 6+25+\left(15-10\right)i.

$$Re(\frac{31+5i}{29})$$

Divide 31+5i by 29 to get \frac{31}{29}+\frac{5}{29}i.

$$Re(\frac{31}{29}+\frac{5}{29}i)$$

The real part of \frac{31}{29}+\frac{5}{29}i is \frac{31}{29}.

$$\frac{31}{29}$$