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**Unformatted text preview: **st property which makes compositions of roots and powers delicate. This is especially
true when we use exponential notation for radicals. Recall the following deﬁnition.
Definition 5.5. Let x be a real number, m an integera and n a natural number.
1 m • xn
a √
n √
x and is deﬁned whenever n x is deﬁned.
√
√m
√m
= ( n x) = n xm , whenever ( n x) is deﬁned. • xn = Recall this means m = 0, ±1, ±2, . . . The rational exponents deﬁned in Deﬁnition 5.5 behave very similarly to the usual integer exponents
3/2
from Elementary Algebra with one critical exception. Consider the expression x2/3
. Applying
3/2 23 the usual laws of exponents, we’d be tempted to simplify this as x2/3
= x 3 · 2 = x1 = x.
√
2
However, if we substitute x = −1 and apply Deﬁnition 5.5, we ﬁnd (−1)2/3 = 3 −1 = (−1)2 = 1
√3
3/2
3/2
= x. If we take
so that (−1)2/3
= 13/2 =
1 = 13 = 1. We see in this case that x2/3
the time to rewrite x2/3
x2/3 3/2 3/2 = with radicals, we see
√
3
x 2 3/2 = √...

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