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Simplifying Expressions with Rational Exponents

Learn how to simplify expressions with rational exponents using inverse operations, indices, and the Product of Powers Property.

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Simplifying Expressions with Rational Exponents

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  1. Rational Exponents 11-EXT Lesson Presentation Holt Algebra 1

  2. Objective Simplify expressions containing rational exponents.

  3. Vocabulary index

  4. There are inverse operations for other powers as well. For example represents a cube root, and it is the inverse of cubing a number. To find , look for three equal factors whose product is 8. Since 2 2 2 = 8. • • =2. 3 3 3 You have seen that taking a square root and squaring are the inverse operations for nonnegative numbers.

  5. Operation Inverse Example In the table, notice the small number to the left of the radical sign. It indicate what root you are taking. This is the index (plural, indices) of the radical. When a radical is written without an index, the index is understood to be 2. Only positive integers greater than or equal to 2 may be indices of radicals.

  6. Think 2 = 196 3 Think 3 = 125 4 Think 4 = 10,000 6 Think 6 = 64 6 Example 1: Simplifying Roots Simplify each expression. A. B. 3 C. 4 D.

  7. 3 Think 3 = 27 5 Think 5 = 0 4 4 Think 4= 16 Think 2= 144 Check It Out! Example 1 Simplify each expression. 3 a. 5 b. c. d.

  8. 2 So, = 3. However, = 3 also. Since squaring either expression gives a result of 3, it must be true that . You have seen that exponents can be integers. Exponents can also be fractions. What does it mean when an exponent is a fraction? For example, what is the meaning of 2 Product of Powers Property. =3

  9. Example 2: Simplifying Think 3 = 343. Think 5 = 32. Simplify each expression A. Use the definition of . 3 = 7 B. 5 Use the definition of . = 2

  10. Think 2 = 121. Think 4 = 81. 2 4 Check It Out! Example 2 Simplify each expression. A. Use the definition of . = 11 B. Use the definition of . = 3

  11. Think 4 = 256 4 Check It Out! Example 2 Simplify each expression. C. Use the definition of . = 4

  12. Think 3 = 8. 2 2 3 Definition of . 2 = (2) 3 You can also have a fractional exponent with a numerator other than one. For example, Product of Powers Property. = 4

  13. 2 5 5 2 4 5 Definition of . Example 3: Simplifying Expressions with Rational Exponents Simplify each expression. A. B. Power of a Power Property. = 243 =25

  14. 3 Definition of . 3 4 Example 3C: Simplifying Expressions with Rational Exponents Simplify the expression. Product of Powers Property.

  15. 4 5 Definition of . 3 2 =(2) =(1) Check It Out! Example 3 Simplify each expression. a. b. Power of a Power Property. = 8 = 1

  16. 3 Definition of . 4 = (3) Check It Out! Example 3c Simplify each expression. Power of a Power Property. = 81

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