Exercise 66 Page 213

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. Denominator:& 6-3i Complex Conjugate:& 6+3iThe 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. 5+i/6-3i*6+3i/6+3i This process is also known as rationalizing the denominator. Doing so will simplify the quotient.

5+i/6-3i*6+3i/6+3i

MultFrac

Multiply fractions

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

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Simplify numerator

Distr

Distribute (6+3i)

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

Distr

Distribute 5

30+15i+i(6+3i)/(6-3i)(6+3i)

Distr

Distribute i

30+15i+6i+3i^2/(6-3i)(6+3i)

IDefPow

i^2=- 1

30+15i+6i+3(- 1)/(6-3i)(6+3i)

MultPosNeg

a(- b)=- a * b

30+15i+6i-3/(6-3i)(6+3i)

AddSubTerms

Add and subtract terms

27+21i/(6-3i)(6+3i)

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Simplify denominator

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

27+21i/36+9

AddTerms

Add terms

27+21i/45

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Simplify

WriteSumFrac

Write as a sum of fractions

27/45+21i/45

MovePartNumRight

a* b/c=a/c* b

27/45+21/45i

ReduceFrac

a/b=.a /9./.b /9.

3/5+21/45i

ReduceFrac

a/b=.a /3./.b /3.

3/5+7/15i