Exercise 59 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:& 2 + isqrt(3) Complex Conjugate:& 2 - isqrt(3)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. 2-isqrt(3)/2+isqrt(3)*2-isqrt(3)/2-isqrt(3) This process is also known as rationalizing the denominator. Doing so will simplify the quotient.

2-isqrt(3)/2+isqrt(3)*2-isqrt(3)/2-isqrt(3)

MultFrac

Multiply fractions

(2-isqrt(3))(2-isqrt(3))/(2+isqrt(3))(2-isqrt(3))

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Simplify numerator

Distr

Distribute (2-isqrt(3))

2(2-isqrt(3))-isqrt(3)(2-isqrt(3))/(2+isqrt(3))(2-isqrt(3))

Distr

Distribute 2

4-2isqrt(3)-isqrt(3)(2-isqrt(3))/(2+isqrt(3))(2-isqrt(3))

Distr

Distribute isqrt(3)

4-2isqrt(3)-(2isqrt(3)-3i^2)/(2+isqrt(3))(2-isqrt(3))

Distr

Distribute -1

4-2isqrt(3)-2isqrt(3)+3i^2/(2+isqrt(3))(2-isqrt(3))

IDefPow

i^2=- 1

4-2isqrt(3)-2isqrt(3)+3(- 1)/(2+isqrt(3))(2-isqrt(3))

MultPosNeg

a(- b)=- a * b

4-2isqrt(3)-2isqrt(3)-3/(2+isqrt(3))(2-isqrt(3))

AddSubTerms

Add and subtract terms

1-4isqrt(3)/(2+isqrt(3))(2-isqrt(3))

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Simplify denominator

ExpandDiffSquares

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

1-4isqrt(3)/4-3i^2

IDefPow

i^2=- 1

1-4isqrt(3)/4-3(-1)

MultNegNegOnePar

- a(- b)=a* b

1-4isqrt(3)/4+3

AddTerms

Add terms

1-4isqrt(3)/7

WriteSumFrac

Write as a sum of fractions

1/7-4isqrt(3)/7