Exercise 58 Page 183

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 it for the denominator of our expression. Denominator:& 4 + isqrt(2) Complex Conjugate:& 4 - isqrt(2)The product of complex conjugates is a real number. To simplify the given expression, we will multiply the numerator and the denominator by the complex conjugate of the denominator. 4-isqrt(2)/4+isqrt(2)*4-isqrt(2)/4-isqrt(2) This process is also known as rationalizing the denominator. Doing so will simplify the quotient.

4-isqrt(2)/4+isqrt(2)*4-isqrt(2)/4-isqrt(2)

MultFrac

Multiply fractions

(4-isqrt(2))(4-isqrt(2))/(4+isqrt(2))(4-isqrt(2))

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Simplify numerator

Distr

Distribute (4-isqrt(2))

4(4-isqrt(2))-isqrt(2)(4-isqrt(2))/(4+isqrt(2))(4-isqrt(2))

Distr

Distribute 4

16-4isqrt(2)-isqrt(2)(4-isqrt(2))/(4+isqrt(2))(4-isqrt(2))

Distr

Distribute ( - isqrt(2))

16-4isqrt(2)-4isqrt(2)+2i^2/(4+isqrt(2))(4-isqrt(2))

IDefPow

i^2=- 1

16-4isqrt(2)-4isqrt(2)+2(- 1)/(4+isqrt(2))(4-isqrt(2))

MultPosNeg

a(- b)=- a * b

16-4isqrt(2)-4isqrt(2)-2/(4+isqrt(2))(4-isqrt(2))

AddSubTerms

Add and subtract terms

14-8isqrt(2)/(4+isqrt(2))(4-isqrt(2))

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Simplify denominator

ExpandDiffSquares

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

14-8isqrt(2)/16-2i^2

IDefPow

i^2=- 1

14-8isqrt(2)/16-2(-1)

MultNegNegOnePar

- a(- b)=a* b

14-8isqrt(2)/16+2

AddTerms

Add terms

14-8isqrt(2)/18

ReduceFrac

a/b=.a /2./.b /2.

7-4isqrt(2)/9

WriteDiffFrac

Write as a difference of fractions

7/9-4isqrt(2)/9

CommutativePropMult

Commutative Property of Multiplication

7/9-4sqrt(2)i/9

MovePartNumRight

a* b/c=a/c* b

7/9-4sqrt(2)/9i