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We’ll use Big mode to make the formulas more readable.
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V 2 + 2
1: (2 + sqrt(2)) / (1 + sqrt(2)) 1: ---------
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V 2 + 1
.
' (2+sqrt(2)) / (1+sqrt(2)) RET d B
Multiplying by the conjugate helps because ‘(a+b) (a-b) = a^2 - b^2’.
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1: (2 + V 2 ) (V 2 - 1)
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a r a/(b+c) := a*(b-c) / (b^2-c^2) RET
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1: V 2
.
a r a*(b+c) := a*b + a*c
(We could have used a x instead of a rewrite rule for the second step.)
The multiply-by-conjugate rule turns out to be useful in many
different circumstances, such as when the denominator involves
sines and cosines or the imaginary constant i.