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**Unformatted text preview: **3 x 2 3 = √
3 x 3 = √
3 x 3 = |x| In the play-by-play analysis, we see that when we canceled the 2’s in multiplying
2 Otherwise we’d run into the same paradox we did in Section 3.4. 2
3 · 3 , we were,
2 5.3 Other Algebraic Functions 313 in fact, attempting to cancel a square with a square root. The fact that √
x2 = |x| and not
2/3 simply x is the root3 of the trouble. It may amuse the reader to know that x3/2
= x, and
this veriﬁcation is left as an exercise. The moral of the story is that when simplifying fractional
exponents, it’s usually best to rewrite them as radicals.4 The last major property we will state,
and leave to Calculus to prove, is that radical functions are continuous on their domains, so the
Intermediate Value Theorem, Theorem 3.1, applies. This means that if we take combinations of
radical functions with polynomial and rational functions to form what the authors consider the
algebraic functions,5 we can make sign diagrams using the procedure set forth in Section 4.2.
Steps...

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