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Understanding Exponents and Roots Lesson

Learn to simplify expressions, apply fractional exponents, and understand root operations in this warm-up lesson presentation. Explore examples and practice exercises to master the concepts.

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Understanding Exponents and Roots Lesson

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  1. Preview Warm Up California Standards Lesson Presentation

  2. Warm Up Simplify each expression. 1. 2. 3. 4. 5. 6. 6 0 4 1 10 –3

  3. 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. California Standards

  4. Vocabulary index

  5. Recall that the radical symbol is used to indicate roots. The indexis the small number to the left of the radical symbol that tells which root to take. For example represents a cubic root. Since 23 = 222 = 8,

  6. Another way to write nth roots is by using fractional exponents. For example, for b >1, suppose Square both sides. b1 = b2k Power of a Power Property 1 = 2k If bm = bn, then m = n. Divide both sides by 2. So for all b > 1,

  7. Helpful Hint When b = 0, When b = 1,

  8. 1 1 1 n n n Use the definition of . b b Use the definition of . Additional Example 1: Simplifying b Simplify each expression. A. = 7 B. = 2 + 3 = 5

  9. 1 1 n n Use the definition of . b b Use the definition of . Check It Out! Example 1 Simplify each expression. a. = 3 b. = 11 + 4 = 15

  10. Definition of A fractional exponent can have a numerator other than 1, as in the expression . You can write the exponent as a product in two different ways. Power of a Power Property

  11. Definition of = 243 = 25 Additional Example 2: Simplifying Expressions with Fractional Exponents Simplify each expression. A. B.

  12. Definition of Check It Out! Example 2 Simplify each expression. a. b. = (1)3 = 1 = 8

  13. Definition of Check It Out! Example 2 Simplify each expression. c. = 81

  14. Definition of Additional Example 3: Application Given a cube with surface area S, the volume V of the cube can be found by using the formula Find the volume of a cube with surface area 54 m2. Substitute 54 for s. Simplify inside the parentheses. The volume of the cube is 27 m3.

  15. Definition of Check It Out! Example 3 The approximate number of Calories C that an 2 animal needs each day is given by , where m is the animal’s mass in kilograms. Find the number of Calories that an 81 kg panda needs each day. Substitute 81 for m. The panda needs 1944 Calories per day to maintain health. = 7227 = 1944

  16. Remember that always indicates a nonnegative square root. When you simplify variable expressions that contain , such as , the answer cannot be negative. But x may be negative. Therefore you simplify as |x| to ensure the answer is nonnegative.

  17. When n is even, you must simplify to |x|, because you do not know whether x is positive or negative. When n is odd, simplify to x.

  18. Helpful Hint When you are told that all variables represent nonnegative numbers, you do not need to use absolute values in your answer.

  19. • Definition of Additional Example 4A: Properties of Exponents to Simplify Expressions Simplify. All variables represent nonnegative numbers. Power of a Product Property Power of a Power Property Simplify exponents.

  20. • • Additional Example 4B: Properties of Exponents to Simplify Expressions Simplify. All variables represent nonnegative numbers. Power of a Product Property Simplify exponents. Product of Powers Property

  21. Definition of Check It Out! Example 4a Simplify. All variables represent nonnegative numbers. Power of a Product Property Simplify exponents.

  22. Power of a Product Property and Check It Out! Example 4b Simplify. All variables represent nonnegative numbers. Simplify. = xy

  23. Lesson Quiz: Part I Simplify each expression. 1. 2. 3. 4. 2 9 729 128

  24. Lesson Quiz: Part II 5. In an experiment, the approximate population P of a bacteria colony is given by , where t is the number of days since the start of the experiment. Find the population of the colony on the 8th day. 480 Simplify. All variables represent nonnegative numbers. 6. 7.

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