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11.1.1 Definition of a square root
When you take the square root of a number, you’re looking for the number that if you square
you get back. Hmmm…that’s not very clear. Let’s try: because 32 = 9. You might also
think of it as . The square root and the square (little 2) on the 3 cancel. We can only take the square
roots of non-negative numbers. (We’ll learn what to do with negative numbers in 11.8.)
If there’s a negative sign out in front of a square root (or any other root for that matter), do the
root first, then apply the negative sign. In the order of operations, roots are on par with exponents.
11.1.2 Definition of an nth root
The radical expression has 3 parts…the radicand (the part underneath the symbol), the index (the
little number outside to the left of the symbol), and the power the radicand is raised to.
The index is the root. If the index is 2, it’s a square root, and we don’t typically write the
2. All other indices we write.
So, in the expression
a is the radicand
n is the index
m is the power
If the index is an odd number, the radicand can be any real number.
, since (-2)3 = -8.
11.1.3 Roots of Variable Expressions
If there’s a variable radicand, and you can write it as an amount raised to a power equal to the
index (or a multiple of the index), you can simplify the radical.
EX)
NOTE: The book (really all Intermediate Algebra textbooks) make a big deal in this section (and
only in this section) about: If the index is an even number, the outcome has to be a positive
amount. With numbers, this is no big deal, you just make it positive.
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