Solve 8+9i/3-i

$$\frac { 8 + 9 i } { 3 - i }$$

Evaluate

$$\frac{3}{2}+\frac{7}{2}i=1.5+3.5i$$

Solution Steps

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

$$\frac{\left(8+9i\right)\left(3+i\right)}{\left(3-i\right)\left(3+i\right)}$$

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

$$\frac{\left(8+9i\right)\left(3+i\right)}{3^{2}-i^{2}}$$

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

$$\frac{\left(8+9i\right)\left(3+i\right)}{10}$$

Multiply complex numbers 8+9i and 3+i like you multiply binomials.

$$\frac{8\times 3+8i+9i\times 3+9i^{2}}{10}$$

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

$$\frac{8\times 3+8i+9i\times 3+9\left(-1\right)}{10}$$

Do the multiplications in 8\times 3+8i+9i\times 3+9\left(-1\right).

$$\frac{24+8i+27i-9}{10}$$

Combine the real and imaginary parts in 24+8i+27i-9.

$$\frac{24-9+\left(8+27\right)i}{10}$$

Do the additions in 24-9+\left(8+27\right)i.

$$\frac{15+35i}{10}$$

Divide 15+35i by 10 to get \frac{3}{2}+\frac{7}{2}i.

$$\frac{3}{2}+\frac{7}{2}i$$

Real Part

$$\frac{3}{2} = 1\frac{1}{2} = 1.5$$

Solution Steps

Multiply both numerator and denominator of \frac{8+9i}{3-i} by the complex conjugate of the denominator, 3+i.

$$Re(\frac{\left(8+9i\right)\left(3+i\right)}{\left(3-i\right)\left(3+i\right)})$$

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

$$Re(\frac{\left(8+9i\right)\left(3+i\right)}{3^{2}-i^{2}})$$

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

$$Re(\frac{\left(8+9i\right)\left(3+i\right)}{10})$$

Multiply complex numbers 8+9i and 3+i like you multiply binomials.

$$Re(\frac{8\times 3+8i+9i\times 3+9i^{2}}{10})$$

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

$$Re(\frac{8\times 3+8i+9i\times 3+9\left(-1\right)}{10})$$

Do the multiplications in 8\times 3+8i+9i\times 3+9\left(-1\right).

$$Re(\frac{24+8i+27i-9}{10})$$

Combine the real and imaginary parts in 24+8i+27i-9.

$$Re(\frac{24-9+\left(8+27\right)i}{10})$$

Do the additions in 24-9+\left(8+27\right)i.

$$Re(\frac{15+35i}{10})$$

Divide 15+35i by 10 to get \frac{3}{2}+\frac{7}{2}i.

$$Re(\frac{3}{2}+\frac{7}{2}i)$$

The real part of \frac{3}{2}+\frac{7}{2}i is \frac{3}{2}.

$$\frac{3}{2}$$