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Friday 1st of July 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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Exponential Notation
If a is any real number and n is a positive integer, then the nth power of a is The number a is called the base and n is called the exponent.

Zero and Negative Exponents
If a ≠ 0 is any real number and n is a positive integer, then Laws of Exponents Example 1. Evaluate the expressions Example 2. Simplify the expressions Solution. law 4 and 3 rearrange factors factor 6 and use law 1  definition of negative exponents means b2 = a and b ≥ 0.

Example. ,
since 32 = 9 and 3 ≥0.
Definition of nth Root
If n > 1 is a positive integer, then the principal nth root of a, denoted by is
(1) 0 if a = 0.
(2) the positive number b such that bn = a, if a is positive.
(3) (a) the negative number b such that bn = a, if n is odd and a is negative.
(b) not a real number if n is even and a is positive.

If n is even, we must have a ≥ 0 and b ≥ 0. Complex numbers are needed to define if
a < 0 and n is an even positive integer, because when n is even for all real numbers b, we
have bn ≥ 0.
If n = 2 we write instead of .
Example. since 33 = 81 and 3 ≥ 0.
Example. since (-2)3 = -8.
Note that are not defined as real numbers.

Properties of nth Roots Note that we have . For example, Rational Exponents
Definition. If m and n are integers and n > 1, and if a is a real number such that exists, we define or equivalently Note that by the definition Remark. The laws of exponents hold for rational exponents also.
Example Evaluate the expressions Example Simplify the expressions Example Simplify the expression and eliminate any negative exponent(s). Rationalizing the Denominator
Rationalizing the denominator is to eliminate the radical in a denominator by multiplying
both numerator and denominator by an appropriate expression.
Example. In general, if the denominator is of the form n with m < n, then multiplying the
numerator and denominator by Example Rationalize the denominator More Examples. Simplify the expressions and eliminate any negative exponent(s) 