Exercise 23 Page 188

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. cc Denominator:& 2+5i Complex Conjugate:& 2-5iThe 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. 3-i/2+5i*2-5i/2-5i This process is also known as rationalizing the denominator. Doing so will simplify the quotient.

3-i/2+5i * 2-5i/2-5i

MultFrac

Multiply fractions

(3-i)(2-5i)/(2+5i)(2-5i)

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Simplify numerator

Distr

Distribute 3-i

(3-i)2-(3-i)5i/(2+5i)(2-5i)

Distr

Distribute 2

6-2i-(3-i)5i/(2+5i)(2-5i)

Distr

Distribute 5i

6-2i-(15i-5i^2)/(2+5i)(2-5i)

DistrNegSignSwap

-(b-a)=a-b

6-2i+5i^2-15i/(2+5i)(2-5i)

IDefPow

i^2=- 1

6-2i+5(- 1)-15i/(2+5i)(2-5i)

MultPosNeg

a(- b)=- a * b

6-2i-5-15i/(2+5i)(2-5i)

SubTerms

Subtract terms

1-17i/(2+5i)(2-5i)

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Simplify denominator

Distr

Distribute 2-5i

1-17i/(2-5i)2+(2-5i)5i

Distr

Distribute 2

1-17i/4-10i+(2-5i)5i

Distr

Distribute 5i

1-17i/4-10i+10i-25i^2

IDefPow

i^2=- 1

1-17i/4-10i+10i-25(- 1)

MultNegNegOnePar

- a(- b)=a* b

1-17i/4-10i+10i+25

AddSubTerms

Add and subtract terms

1-17i/29

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Simplify

WriteDiffFrac

Write as a difference of fractions

1/29-17i/29

MovePartNumRight

a* b/c=a/c* b

1/29-17/29i