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**Unformatted text preview: **f the basic properties of radicals from Intermediate Algebra.1
Definition 5.4. Let x be a real number and n a natural number.a If n is odd, the principal
√
√n
√
nth root of x, denoted n x is the unique real number satisfying ( n x) = x. If n is even, n x is
√
deﬁned similarlyb provided x ≥ 0 and n x ≥ 0. The index is the number n and the radicand is
√
√
the number x. For n = 2, we write x instead of 2 x.
a
b Recall this means n = 1, 2, 3, . . ..
√
Recall both x = −2 and x = 2 satisfy x4 = 16, but 4 16 = 2, not −2. √
It is worth remarking that, in light of Section 5.2, we could deﬁne f (x) = n x functionally as the
inverse of g (x) = xn with the stipulation that when n is even, the domain of g is restricted to [0, ∞).
From what we know about g (x) = xn from Section 3.1 along with Theorem 5.3, we can produce
√
the graphs of f (x) = n x by reﬂecting the graphs of g (x) = xn across the line y = x. Below are the
√
√
√
graphs of y = x, y = 4 x and y = 6 x. The point (0, 0) is indicated as a reference. The axes are
hidden so we can see...

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