Exercise 97 Page 373

Recall that the number pairs $a+bi$ and $a-bi$ are complex conjugates. To write the complex conjugate of a complex number, we only change the sign of the imaginary part. Let's do that for the denominator of our given expression now. The product of complex conjugates is a real number. To write the given quotient as a complex number, we will multiply the numerator and the denominator by the complex conjugate of the denominator.

$$\begin{array}{c}\frac{4}{4+i}\cdot \frac{4-i}{4-i}\end{array}$$

This process is also known as rationalizing the denominator. Doing so will simplify the quotient.

$\frac{4}{4+i}\cdot \frac{4-i}{4-i}$

MultFrac

Multiply fractions

$\frac{4(4-i)}{(4+i)(4-i)}$

Distr

Distribute $4$

$\frac{16-4i}{(4+i)(4-i)}$

▼

Simplify denominator

ExpandDiffSquares

$(a+b)(a-b)=a^2-b^2$

$\frac{16-4i}{16-i^2}$

IDefPow

$i^2=\text{-}1$

$\frac{16-4i}{16-(\text{-}1)}$

SubNeg

$a-(\text{-}b)=a+b$

$\frac{16-4i}{16+1}$

AddTerms

Add terms

$\frac{16-4i}{17}$

WriteDiffFrac

Write as a difference of fractions

$\frac{16}{17}-\frac{4i}{17}$

MovePartNumRight

$\frac{a\cdot b}{c}=\frac{a}{c}\cdot b$

$\frac{16}{17}-\frac{4}{17}i$