Solve (6-7i)div(4-5i)=

$$( 6 - 7 i ) \div ( 4 - 5 i ) =$$

Evaluate

$$\frac{59}{41}+\frac{2}{41}i\approx 1.43902439+0.048780488i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 4+5i.

$$\frac{\left(6-7i\right)\left(4+5i\right)}{\left(4-5i\right)\left(4+5i\right)}$$

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

$$\frac{\left(6-7i\right)\left(4+5i\right)}{4^{2}-5^{2}i^{2}}$$

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

$$\frac{\left(6-7i\right)\left(4+5i\right)}{41}$$

Multiply complex numbers 6-7i and 4+5i like you multiply binomials.

$$\frac{6\times 4+6\times \left(5i\right)-7i\times 4-7\times 5i^{2}}{41}$$

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

$$\frac{6\times 4+6\times \left(5i\right)-7i\times 4-7\times 5\left(-1\right)}{41}$$

Do the multiplications in 6\times 4+6\times \left(5i\right)-7i\times 4-7\times 5\left(-1\right).

$$\frac{24+30i-28i+35}{41}$$

Combine the real and imaginary parts in 24+30i-28i+35.

$$\frac{24+35+\left(30-28\right)i}{41}$$

Do the additions in 24+35+\left(30-28\right)i.

$$\frac{59+2i}{41}$$

Divide 59+2i by 41 to get \frac{59}{41}+\frac{2}{41}i.

$$\frac{59}{41}+\frac{2}{41}i$$

Real Part

$$\frac{59}{41} = 1\frac{18}{41} = 1.4390243902439024$$

Solution Steps

Multiply both numerator and denominator of \frac{6-7i}{4-5i} by the complex conjugate of the denominator, 4+5i.

$$Re(\frac{\left(6-7i\right)\left(4+5i\right)}{\left(4-5i\right)\left(4+5i\right)})$$

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

$$Re(\frac{\left(6-7i\right)\left(4+5i\right)}{4^{2}-5^{2}i^{2}})$$

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

$$Re(\frac{\left(6-7i\right)\left(4+5i\right)}{41})$$

Multiply complex numbers 6-7i and 4+5i like you multiply binomials.

$$Re(\frac{6\times 4+6\times \left(5i\right)-7i\times 4-7\times 5i^{2}}{41})$$

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

$$Re(\frac{6\times 4+6\times \left(5i\right)-7i\times 4-7\times 5\left(-1\right)}{41})$$

Do the multiplications in 6\times 4+6\times \left(5i\right)-7i\times 4-7\times 5\left(-1\right).

$$Re(\frac{24+30i-28i+35}{41})$$

Combine the real and imaginary parts in 24+30i-28i+35.

$$Re(\frac{24+35+\left(30-28\right)i}{41})$$

Do the additions in 24+35+\left(30-28\right)i.

$$Re(\frac{59+2i}{41})$$

Divide 59+2i by 41 to get \frac{59}{41}+\frac{2}{41}i.

$$Re(\frac{59}{41}+\frac{2}{41}i)$$

The real part of \frac{59}{41}+\frac{2}{41}i is \frac{59}{41}.

$$\frac{59}{41}$$