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Radicals

The Basics

(0) In this discussion we will ONLY deal with REAL quantities – that is, no Complex Numbers

(1) The index for any radical can only be a positive integer greater than 1. For such indices, we make these definitions:

If n is odd,

If n is even and , x ≥ 0

where b ≥ 0 and

For odd indices can be positive, zero or negative and the resulting will be positive, zero or negative respectively. For example, However, for even indices the radicand MUST be non-negative and the resulting will also be non-negative. For example, but is not defined. (Actually, it can be defined, but it involves i, and we’ve agreed not to use i here.)

(4) We have another notation for that uses exponents: These fractional exponents obey all the laws of exponents. For example, “when you multiply the bases, you add the exponents”: However, we haven’t defined objects like yet so we won’t push this idea here. So far we’ve only defined the meaning of fractional exponents if the numerator is 1 (a unit fraction).

(5) On your calculator you use parentheses around the exponent. For example, is entered on the calculator as . Similarly, would be entered as

(6) We have the following rule.

If n is odd,

If n is even and

These rules make sense if you look at the exponent form: Thus, you can eliminate the radical if the radicand is an object raised to a power which is a multiple of the index. For example,

(7) Note: This is obvious if x is non-negative: It really comes into play if x is negative: which is NOT what we started with. In this case which is what the rule requires.

Some Manipulative Skills

(1) A basic definition:

If m and n are integers with then

provided is real.

We have the option on to (a) root first, then power or (b) power first, then root. However, the “root first, then power” part of the definition is the ONLY WAY we want to do these things. It is in keeping with the idea of make small before making big that we saw long ago. So, for example:

(2) Other manipulations:

	mult. & div. only -> put power throughthe product
	divide the bases -> subtract the exponents

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

	mult. & div. only -> flip LAST
	divide the basis -> subtract the exponents
	now flip

	indices different -> convert to fractional exponents

(3) More rules:

provided at least one of and is real

Remember:

There is NO RULE for simplifying a radicand that is a sum or difference:

We combine the rules above with the following rule to not only manipulate but also simplify radicals:

(S)

provided if n is even

For example, to simplify we begin by finding the largest multiples of 3 that do not exceed the given exponents: 3, 12, and 9 respectively in this case. We “peel these powers off” Now, we can apply (S) along with (M1) all at once after we note that and