Solve 7+3i/8-9i

$$\frac { 7 + 3 i } { 8 - 9 i }$$

Evaluate

$$\frac{1}{5}+\frac{3}{5}i=0.2+0.6i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 8+9i.

$$\frac{\left(7+3i\right)\left(8+9i\right)}{\left(8-9i\right)\left(8+9i\right)}$$

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

$$\frac{\left(7+3i\right)\left(8+9i\right)}{8^{2}-9^{2}i^{2}}$$

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

$$\frac{\left(7+3i\right)\left(8+9i\right)}{145}$$

Multiply complex numbers 7+3i and 8+9i like you multiply binomials.

$$\frac{7\times 8+7\times \left(9i\right)+3i\times 8+3\times 9i^{2}}{145}$$

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

$$\frac{7\times 8+7\times \left(9i\right)+3i\times 8+3\times 9\left(-1\right)}{145}$$

Do the multiplications in 7\times 8+7\times \left(9i\right)+3i\times 8+3\times 9\left(-1\right).

$$\frac{56+63i+24i-27}{145}$$

Combine the real and imaginary parts in 56+63i+24i-27.

$$\frac{56-27+\left(63+24\right)i}{145}$$

Do the additions in 56-27+\left(63+24\right)i.

$$\frac{29+87i}{145}$$

Divide 29+87i by 145 to get \frac{1}{5}+\frac{3}{5}i.

$$\frac{1}{5}+\frac{3}{5}i$$

Real Part

$$\frac{1}{5} = 0.2$$

Solution Steps

Multiply both numerator and denominator of \frac{7+3i}{8-9i} by the complex conjugate of the denominator, 8+9i.

$$Re(\frac{\left(7+3i\right)\left(8+9i\right)}{\left(8-9i\right)\left(8+9i\right)})$$

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

$$Re(\frac{\left(7+3i\right)\left(8+9i\right)}{8^{2}-9^{2}i^{2}})$$

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

$$Re(\frac{\left(7+3i\right)\left(8+9i\right)}{145})$$

Multiply complex numbers 7+3i and 8+9i like you multiply binomials.

$$Re(\frac{7\times 8+7\times \left(9i\right)+3i\times 8+3\times 9i^{2}}{145})$$

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

$$Re(\frac{7\times 8+7\times \left(9i\right)+3i\times 8+3\times 9\left(-1\right)}{145})$$

Do the multiplications in 7\times 8+7\times \left(9i\right)+3i\times 8+3\times 9\left(-1\right).

$$Re(\frac{56+63i+24i-27}{145})$$

Combine the real and imaginary parts in 56+63i+24i-27.

$$Re(\frac{56-27+\left(63+24\right)i}{145})$$

Do the additions in 56-27+\left(63+24\right)i.

$$Re(\frac{29+87i}{145})$$

Divide 29+87i by 145 to get \frac{1}{5}+\frac{3}{5}i.

$$Re(\frac{1}{5}+\frac{3}{5}i)$$

The real part of \frac{1}{5}+\frac{3}{5}i is \frac{1}{5}.

$$\frac{1}{5}$$