1. Rational Exponents and Radicals
Section 10.5
Rational Exponents and Radicals

2. 10.5 Lecture Guide: Rational Exponents and Radicals
Objective 1: Interpret and use rational exponents.

3. The principal nth root of the real number x is denoted by either
Examples in
Radical Notation or
Examples in
Exponential Notation
Verbally
For
The principal nth root is positive for all natural numbers n.
For
The principal nth root of 0 is 0.
For
If n is odd the principal nth root is negative.

4. The principal nth root of the real number x is denoted by either
Examples in
Radical Notation or
Examples in
Exponential Notation
Verbally
For
If n is even, there is no real nth root. (The nth roots will be imaginary.)
is not
is not a
a real number.
real number.

5. Use the product rule for exponents to simplify the following expressions:1.
is the principal _________________________
root of x.

6. Use the product rule for exponents to simplify the following expressions:2.
is the principal _________________________
root of x.

7. Use the product rule for exponents to simplify the following expressions:3.
is the principal _________________________
root of x.

8. Represent each expression by using exponential notation, and
evaluate each expression. 4.

9. Represent each expression by using exponential notation, and
evaluate each expression. 5.

10. Represent each expression by using exponential notation, and
evaluate each expression. 6.

11. Represent each expression by using exponential notation, and
evaluate each expression. 7.

12. Represent each expression by using radical notation, and
evaluate each expression. 8.

13. Represent each expression by using radical notation, and
evaluate each expression. 9.

14. Represent each expression by using radical notation, and
evaluate each expression.
10.

15. Represent each expression by using radical notation, and
evaluate each expression.
11.

16. Represent each expression by using radical notation, and
evaluate each expression.
12.

17. Represent each expression by using radical notation, and
evaluate each expression.
13.

18. For a real number x and natural numbers m and n :
Rational Exponents
For a real number x and natural numbers m and n :
Examples in
Radical Notation
Examples in
Exponential Notation
Algebraically If
is a real number* or

19. For a real number x and natural numbers m and n :
Rational Exponents
For a real number x and natural numbers m and n :
Examples in
Radical Notation
Examples in
Exponential Notation
Algebraically If
is a real number*
*If
and n is even, then
is not a real number.

20. Write each expression in radical notation and evaluate.
14.

21. Write each expression in radical notation and evaluate.
15.

22. Write each expression in radical notation and evaluate.
16.

23. Write each expression in radical notation and evaluate.
17.

24. Write each expression in radical notation and evaluate.
18.

25. Write each expression in radical notation and evaluate.
19.

26. Objective 2: Use the properties of exponents.

27. You should be familiar with the properties of integer exponents from Chapter 5. Note that these properties apply to all rational exponents.
Properties of Exponents
Let m and n be real numbers and x, xm, xn ,y, ym, and yn be nonzero real numbers.
Product rule:
Product to a Power:
Quotient rule:
Power rule:
Quotient to a Power:
Negative power:

28. Simplify each expression. Assume that x is a positive real number.
20.

29. Simplify each expression. Assume that x is a positive real number.
21.

30. Simplify each expression. Assume that x is a positive real number.
22.

31. Simplify each expression. Assume that x is a positive real number.
23.

32. Simplify each expression. Assume that x is a positive real number.
24.

33. Simplify each expression. Assume that x and y are positive real numbers.
25.

34. Simplify each expression. Assume that x and y are positive real numbers.
26.

35. Simplify each expression. Assume that x and y are positive real numbers.
27.

36. Simplify each expression. Assume that x and y are positive real numbers.
28.

37. Simplify each expression. Assume that x and y are positive real numbers.
29.

38. Simplify each expression. Assume that x and y are positive real numbers.
30.

39. Simplify each expression. Assume that x and y are positive real numbers.
31.

40. Simplify each expression.
32.

41. Simplify each expression.
33.

42. Simplify each expression.
34.

43. Simplify each expression.
35.

44. Use a graphing calculator or a spreadsheet to approximate
each expression to the nearest hundredth.
36.
37.
38.