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11-EXT 11-EXTRational RationalExponents Exponents Lesson Presentation Holt Algebra Holt Algebra 11 11-EXT Rational Exponents Objective Simplify expressions containing rational exponents. Holt Algebra 1 11-EXT Rational Exponents Vocabulary index Holt Algebra 1 11-EXT Rational Exponents You have seen that taking a square root and squaring are the inverse operations for nonnegative numbers. There are inverse operations for other powers as well. For example 3 represents a cube root, and it is the inverse of cubing a number. To find 3 , look for three equal factors whose product is 8. Since 2 • 2 • 2 = 8. 3 =2. Holt Algebra 1 11-EXT Rational Exponents 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. Holt Algebra 1 11-EXT Rational Exponents Example 1: Simplifying Roots Simplify each expression. A. B. 196 Think 3= 125 Think 4= 10,000 Think 6= 64 4 4 D. 2= 3 3 C. Think 6 6 Holt Algebra 1 11-EXT Rational Exponents Check It Out! Example 1 Simplify each expression. a. 3 3 b. Think 3= 27 Think 5= 0 Think 4= 16 Think 2= 144 5 5 c. 4 4 d. Holt Algebra 1 11-EXT Rational Exponents 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 So, 2 = 3. However, = 3 also. Since squaring either expression gives a result of 3, it must be true that Holt Algebra 1 . 11-EXT Rational Exponents Holt Algebra 1 11-EXT Rational Exponents Example 2: Simplifying Simplify each expression A. 3 =7 Use the definition of Think 3= . 343. B. 5 =2 Holt Algebra 1 Use the definition of Think 5= 32. . 11-EXT Rational Exponents Check It Out! Example 2 Simplify each expression. A. 2 = 11 Use the definition of Think 2= . 121. B. 4 =3 Holt Algebra 1 Use the definition of Think 4= 81. . 11-EXT Rational Exponents Check It Out! Example 2 Simplify each expression. C. 4 =4 Holt Algebra 1 Use the definition of Think 4= 256 . 11-EXT Rational Exponents You can also have a fractional exponent with a numerator other than one. For example, 2 Product of Powers Property. 2 3 Definition of 2 = (2) =4 Holt Algebra 1 Think 3= 8. . 3 11-EXT Rational Exponents Example 3: Simplifying Expressions with Rational Exponents Simplify each expression. A. B. 5 4 = 243 Holt Algebra 1 5 2 Power of a Power Property. Definition of . 5 =25 2 11-EXT Rational Exponents Example 3C: Simplifying Expressions with Rational Exponents Simplify the expression. 3 4 Product of Powers Property. 3 Definition of Holt Algebra 1 . 11-EXT Rational Exponents Check It Out! Example 3 Simplify each expression. a. b. Power of a Power Property. 4 3 =(2) =8 Holt Algebra 1 Definition of . 5 =(1)2 =1 11-EXT Rational Exponents Check It Out! Example 3c Simplify each expression. Power of a Power Property. 3 = (3) 4 = 81 Holt Algebra 1 Definition of .