Solve 8+i/8-i

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

Evaluate

$$\frac{63}{65}+\frac{16}{65}i\approx 0.969230769+0.246153846i$$

Solution Steps

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

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

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

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

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

$$\frac{\left(8+i\right)\left(8+i\right)}{65}$$

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

$$\frac{8\times 8+8i+8i+i^{2}}{65}$$

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

$$\frac{8\times 8+8i+8i-1}{65}$$

Do the multiplications in 8\times 8+8i+8i-1.

$$\frac{64+8i+8i-1}{65}$$

Combine the real and imaginary parts in 64+8i+8i-1.

$$\frac{64-1+\left(8+8\right)i}{65}$$

Do the additions in 64-1+\left(8+8\right)i.

$$\frac{63+16i}{65}$$

Divide 63+16i by 65 to get \frac{63}{65}+\frac{16}{65}i.

$$\frac{63}{65}+\frac{16}{65}i$$

Real Part

$$\frac{63}{65} = 0.9692307692307692$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(8+i\right)\left(8+i\right)}{65})$$

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

$$Re(\frac{8\times 8+8i+8i+i^{2}}{65})$$

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

$$Re(\frac{8\times 8+8i+8i-1}{65})$$

Do the multiplications in 8\times 8+8i+8i-1.

$$Re(\frac{64+8i+8i-1}{65})$$

Combine the real and imaginary parts in 64+8i+8i-1.

$$Re(\frac{64-1+\left(8+8\right)i}{65})$$

Do the additions in 64-1+\left(8+8\right)i.

$$Re(\frac{63+16i}{65})$$

Divide 63+16i by 65 to get \frac{63}{65}+\frac{16}{65}i.

$$Re(\frac{63}{65}+\frac{16}{65}i)$$

The real part of \frac{63}{65}+\frac{16}{65}i is \frac{63}{65}.

$$\frac{63}{65}$$