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Sunday 2nd of October Home Square Roots and Radical Expressions Solving Radical Equations Simplifying Radical Expressions Irrational Numbers in General and Square Roots in Particular Roots of Polynomials Simplifying Radical Expressions Exponents and Radicals Products and Quotients Involving Radicals Roots of Quadratic Equations Radical Expressions Radicals and Rational Exponents Find Square Roots and Compare Real Numbers Radicals Radicals and Rational Exponents Theorems on the Roots of Polynomial Equations SYNTHETIC DIVISION AND BOUNDS ON ROOTS Simplifying Radical Expressions Exponents and Radicals Properties of Exponents and Square Roots Solving Radical Equations Rational Exponents and Radicals,Rationalizing Denominators Rational Exponents and Radicals,Rationalizing Denominators Quadratic Roots Exponents and Roots Multiplying Radical Expressions Exponents and Radicals Solving Radical Equations Solving Quadratic Equations by Factoring and Extracting Roots Newton's Method for Finding Roots Roots of Quadratic Equations Studio Roots, Radicals, and Root Functions Review division factoring and Root Finding Radicals Simplifying Radical Expressions Multiplying and Simplifying Radical Expressions LIKE RADICALS Multiplication and Division of Radicals Radical Equations BOUNDING ROOTS OF POLYNOMIALS
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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 