56 = 15,625 so 15,6251/6= 5
− 37 =−2,187 so (−2,187)1/7=− 3
(−5)^9 =−1,953,125, so (−1,953,125)1/9=− 5
64 = 1,296 so 1,2961/4= 6
(−6)^4 = 1,296 so 1,2961/4=−6 ... What?
The radical notation can be used for any integer root. For the 1/n power, a small numeral n is
placed in the upper left part of the radical symbol. If you use this notation, you must be sure
that the radical symbol extends completely over the quantity of which you want to take the
root. If you use the fractional notation, parentheses, brackets, and braces should be used to
define the quantity of which you want to take the root.
Are you confused?
Now you will ask, “Can 6 and −6 both be valid 4th roots of 1,296?” The answer is “Yes.” Both 6 and − 6
will work here:
6 × 6 × 6 × 6 = 1,296and
(−6) × (−6) × (−6) × (−6)= 1,296If you multiply any negative number by itself an even number of times, you’ll get a positive number.
Therefore, if you have some number a and its additive inverse −a, and then you raise both of those num-
bers to an even positive integer power p, you will get
(−a)p=apevery time! If we call (−a)p or ap by some other name such as b, then the pth root of b is ambiguous. That
would mean, for example,
16 1/4= 2 and − 2
81 1/4= 3 and − 3
15,6251/6= 5 and − 5
It could even mean something as simple, and yet as troubling, as
1 1/2= 1 and − 1Mathematicians get around this problem by saying that whenever “two numbers at once” are the result of
a reciprocal power, the positive value is the correct one, unless otherwise specified. That means
16 1/4= 2
81 1/4= 3
Reciprocal-of-Integer Powers 115