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

2. Objective
Simplify expressions containing rational exponents.

3. Vocabulary
index

4. 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 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 = 8. 3
=2. •

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

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

8. Product of Powers Property.
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, = 3. However, = 3 also. Since squaring either expression gives a result of 3, it must be true that 2

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

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

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

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

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

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

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

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