6.1 Evaluate Roots and Rational Exponents

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1 VOCABULARY:. Evaluate Roots and Rational Exponents Radical: We know radicals as square roots. But really, radicals can be used to express any root: 0 8, 8, Index: The index tells us exactly what type of root that it is. To determine whether we are looking for pairs in a square root, or sets of three of something in a cube root, the index tells us the size of the group that we are looking to pull out. Rational Exponent: We have a rational exponent when there is a fraction in the exponent position. What we ll find is that we can take a fraction in the exponent position and very easily convert it into a radical. Every radical can be written as a base with a rational exponent. Look at the examples below. What pattern do you notice? 8 8 x x HOW DOES ONE CONVERT FROM RADICAL FORM TO RATIONAL EXPONENT FORM? Take the base under the radical. If it is negative, put it in parenthesis first. The base will then be raised by a fraction created by: The index of the radical becomes the denominator of the fraction. The exponent of the expression becomes the numerator of the fraction. Rewrite the expression using rational exponent notation Rewrite the expression using radical notation

2 Evaluate the expression without a calculator: Radical Form: Convert it to a radical!!. Rewrite as a radical.. Simplify the radical.. Raise your answer from the radical to the exponent. 7 8 Evaluate the expression without a calculator: Evaluate the expression without a calculator: 7 If it s already a radical, then we can start by simplifying!. Simplify the radical.. Raise your answer from the radical to the exponent.

3 To evaluate the expressions with a calculator, you want to first make sure that the expression is written with a rational exponent. Then when you enter in that rational exponent, make sure you put it in as a fraction in parenthesis. As always, all negative bases should be put in parenthesis. Use what we now know to solve an equation. Round the result to two decimal places when appropriate. Isolate the variable or parenthesis containing the variable. Decide what you need to do to both sides to cancel the exponent. Use your calculator to help you solve what remains. Don t forget the +/- when appropriate.. x 7. x 8

4 . Apply Properties of rational exponents Properties of Rational Exponents: Power to another Power, Multiplication, Division, Power to another Power : base stays the same, exponents are multiplied Simplify as much as possible. If possible, put answer in radical form. x y 7 *** 7 x Multiplication: bases must be the same, exponents are added Simplify as much as possible. If possible, put answer in radical form. x x 7 7 or Note: Always rewrite the bigger base!!

5 Division: common bases go immediately to the numerator, exponents are subtracted Simplify as much as possible. If possible, put answer in radical form. 7 7

6 ADDING AND SUBTRACTING RADICALS. Apply Properties of radicals You can add or subtract radicals as long as they have the same index and the same radicand (number underneath the radical). We will call these like radicals because much like like terms on the number in front of the radical will be affected, while the radicand will remain the same. Before you begin any addition or subtraction problem, you should simplify each radical as much as possible. For example: 0 should be simplified to Now, decide whether they are able to be added (same index and same radicand). If yes, only add the number in front of each radical, keeping both the radicand and index the same. x x 7 MULTIPLYING RADICAL BY RADICALS You can multiply a radical by another radical as long as they both have the same index. To multiply a radical by another radical, multiply the numbers in front, multiply the radicands, and keep the index the same. Then check your answer to see if anything can be pulled out. If not, it is in simplified form x x

7 One of the fundamental rules of radicals is that a radical cannot be left in the denominator of a fraction. For example, when given root in the denominator, we are looking for what we can multiply both the numerator and the denominator by in order to create a quantity in the denominator that can be completely pulled out. DIVIDING WITH RADICALS To rationalize a radical is to multiply both the numerator and denominator by a quantity that will, with simplification, convert the denominator from a radical to an integer. We can divide the radicands of two radicals as long as they have the same index. Another example: Another example: Always try to reduce before doing anything else!!!!! y x

8 PROBLEMS WITH RADICALS AND RATIONAL EXPONENTS When one doesn t work, we try the other!! 9 7

9 . Perform Function Operation and Composition Let f(x) = x / 9, g(x) = x / +. Perform the indicated operation and state the domain of the new function. f(x) + g(x) Domain:. Examine the domain of f(x). Is x in the denominator or part of an even root?. Examine the domain of g(x). Is x in the denominator or part of an even root? Domain: Let f(x) = 8x / + and h(x) = x / 9. Perform the indicated operation and state the domain of the new function. h(x) f(x) Domain:. Examine the domain of h(x). Is x in the denominator or part of an even root?. Examine the domain of f(x). Is x in the denominator or part of an even root? Domain: Let f(x) = x, g(x) = x / and h(x) = x /. Perform the indicated operation and state the domain of the new function. f(x) h(x) h ( x ) g ( x ) Domain:. Examine the domain of f(x). Is x in the denominator or part of an even root? Domain:. Examine the domain of h(x). Is x in the denominator or part of an even root?. Examine the domain of h(x). Is x in the denominator or part of an even root?. Examine the domain of g(x). Is x in the denominator or part of an even root? Domain: Domain:

10 COMPOSITION OF FUNCTIONS: Composing one function with another function. Function f has its inputs (the x values) and the outputs (the set of y values often noted f(x)). In a composition, the outputs of one function become the inputs of another function. In this case, the outputs of f will become in the inputs of g. Our job is to find the new function created as a result. We call this a composition and denote it g(f (x)). Let f(x) = x +, f(g(0)) ( ) x h x, and g(x) =. Find the following. x h(g( )) 0 will be input into function g. It s output will become the input of f. Let f(x) = x, g(f(x)) 8 h ( x) x, g(x) = x 7. Find the following. Then, state the domain. h(f(x)) f(g(x))