Solve 6+2i/7-3i

$$\frac { 6 + 2 i } { 7 - 3 i }$$

Evaluate

$$\frac{18}{29}+\frac{16}{29}i\approx 0.620689655+0.551724138i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 7+3i.

$$\frac{\left(6+2i\right)\left(7+3i\right)}{\left(7-3i\right)\left(7+3i\right)}$$

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

$$\frac{\left(6+2i\right)\left(7+3i\right)}{7^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(6+2i\right)\left(7+3i\right)}{58}$$

Multiply complex numbers 6+2i and 7+3i like you multiply binomials.

$$\frac{6\times 7+6\times \left(3i\right)+2i\times 7+2\times 3i^{2}}{58}$$

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

$$\frac{6\times 7+6\times \left(3i\right)+2i\times 7+2\times 3\left(-1\right)}{58}$$

Do the multiplications in 6\times 7+6\times \left(3i\right)+2i\times 7+2\times 3\left(-1\right).

$$\frac{42+18i+14i-6}{58}$$

Combine the real and imaginary parts in 42+18i+14i-6.

$$\frac{42-6+\left(18+14\right)i}{58}$$

Do the additions in 42-6+\left(18+14\right)i.

$$\frac{36+32i}{58}$$

Divide 36+32i by 58 to get \frac{18}{29}+\frac{16}{29}i.

$$\frac{18}{29}+\frac{16}{29}i$$

Real Part

$$\frac{18}{29} = 0.6206896551724138$$

Solution Steps

Multiply both numerator and denominator of \frac{6+2i}{7-3i} by the complex conjugate of the denominator, 7+3i.

$$Re(\frac{\left(6+2i\right)\left(7+3i\right)}{\left(7-3i\right)\left(7+3i\right)})$$

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

$$Re(\frac{\left(6+2i\right)\left(7+3i\right)}{7^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(6+2i\right)\left(7+3i\right)}{58})$$

Multiply complex numbers 6+2i and 7+3i like you multiply binomials.

$$Re(\frac{6\times 7+6\times \left(3i\right)+2i\times 7+2\times 3i^{2}}{58})$$

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

$$Re(\frac{6\times 7+6\times \left(3i\right)+2i\times 7+2\times 3\left(-1\right)}{58})$$

Do the multiplications in 6\times 7+6\times \left(3i\right)+2i\times 7+2\times 3\left(-1\right).

$$Re(\frac{42+18i+14i-6}{58})$$

Combine the real and imaginary parts in 42+18i+14i-6.

$$Re(\frac{42-6+\left(18+14\right)i}{58})$$

Do the additions in 42-6+\left(18+14\right)i.

$$Re(\frac{36+32i}{58})$$

Divide 36+32i by 58 to get \frac{18}{29}+\frac{16}{29}i.

$$Re(\frac{18}{29}+\frac{16}{29}i)$$

The real part of \frac{18}{29}+\frac{16}{29}i is \frac{18}{29}.

$$\frac{18}{29}$$