Exercise 98 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.

$$\begin{array}{c}\frac{\text{-}1}{7-5i}\cdot \frac{7+5i}{7+5i}\end{array}$$

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

$\frac{\text{-}1}{7-5i}\cdot \frac{7+5i}{7+5i}$

MultFrac

Multiply fractions

$\frac{\text{-}1(7+5i)}{(7-5i)(7+5i)}$

Distr

Distribute $\text{-}1$

$\frac{\text{-}7-5i}{(7-5i)(7+5i)}$

▼

Simplify denominator

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

$\frac{\text{-}7-5i}{7^2-(5i{)}^2}$

PowProdII

${\left(ab\right)}^m=a^m{b}^m$

$\frac{\text{-}7-5i}{7^2-5^2{i}^2}$

IDefPow

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

$\frac{\text{-}7-5i}{7^2-5^2(\text{-}1)}$

CalcPow

Calculate power

$\frac{\text{-}7-5i}{49-25(\text{-}1)}$

MultNegNegOnePar

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

$\frac{\text{-}7-5i}{49+25}$

AddTerms

Add terms

$\frac{\text{-}7-5i}{74}$

▼

Simplify

WriteDiffFrac

Write as a difference of fractions

$\frac{\text{-}7}{74}-\frac{5i}{74}$

MovePartNumRight

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

$\frac{\text{-}7}{74}-\frac{5}{74}i$

MoveNegNumToFrac

Put minus sign in front of fraction

$\text{-}\frac{7}{74}-\frac{5}{74}i$