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#### 2.7.52 Rewrites Tutorial Exercise 1

We’ll use Big mode to make the formulas more readable.

                                               ___
                                              V 2  + 2
    1:  (2 + sqrt(2)) / (1 + sqrt(2))     1:  ---------
        .                                      ___
                                              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`’.

              ___    ___
    1:  (2 + V 2 ) (V 2  - 1)
        .

      a r a/(b+c) := a*(b-c) / (b^2-c^2) RET

         ___
    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`.