10.5 Lecture Guide: Rational Exponents and Radicals
Objective 1: Interpret and use rational exponents.
Principal nth Root
For 0 :x
The principal nth root is positive for all natural numbers n.
For 0 :x
The principal nth root of 0 is 0.
For 0 :x
If n is odd the principal nth root is negative.
Examples inRadical Notation
16
3 27
0 0
3 0 0
3 64 5 32
The principal nth root of the real number x is denoted by either 1nx or .n x Examples in
Exponential Notation Verbally
1216
1327
120 0
130 0
1
364
1
532
Principal nth Root
Examples inRadical Notation
The principal nth root of the real number x is denoted by either 1nx or .n x Examples in
Exponential Notation Verbally
If n is even, there is no real nth root. (The nth roots will be imaginary.)
For 0 :x
1
a real number. is not
121
real number. is not a
Use the product rule for exponents to simplify the following expressions:
1.1 1
2 2x x
12x is the principal _________________________ root of x.
Use the product rule for exponents to simplify the following expressions:
2.1 1 1
3 3 3x x x
13x is the principal _________________________ root of x.
Use the product rule for exponents to simplify the following expressions:
3.1 1 1 1
4 4 4 4x x x x
14x is the principal _________________________ root of x.
4. 81
Represent each expression by using exponential notation, and evaluate each expression.
5.
Represent each expression by using exponential notation, and evaluate each expression.
4 81
6.
Represent each expression by using exponential notation, and evaluate each expression.
7 1
7.
Represent each expression by using exponential notation, and evaluate each expression.
3 8
Represent each expression by using radical notation, and evaluate each expression.
8.1
216
9.1
38
Represent each expression by using radical notation, and evaluate each expression.
10.1
416
Represent each expression by using radical notation, and evaluate each expression.
11.1
249
Represent each expression by using radical notation, and evaluate each expression.
12. 1
249
Represent each expression by using radical notation, and evaluate each expression.
13. 1
3125
Represent each expression by using radical notation, and evaluate each expression.
Examples inRadical Notation
Examples in Exponential Notation Algebraically
For a real number x and natural numbers m and n :
If 1nx
is a real number*
1 m mmnn nx x x
or
1m nm mnnx x x
2233
2
8 8
2
4
22 13 3
2
8 8
2
4
Rational Exponents
3377y y
3 137 7y y
Examples inRadical Notation
Examples in Exponential Notation Algebraically
For a real number x and natural numbers m and n :
If 1nx
is a real number*
1mn
mn
xx
3
43
4
3
116
16
1
21
8
34
34
314
3
116
161
16
1 1
2 8
*If 0x and n is even, then 1nx is not a real number.
Rational Exponents
, 0x
Write each expression in radical notation and evaluate.
14.3
216
15.2
38
Write each expression in radical notation and evaluate.
16. 2
38
Write each expression in radical notation and evaluate.
17. 3
481
Write each expression in radical notation and evaluate.
18.2
327
Write each expression in radical notation and evaluate.
19. 1
3125
Write each expression in radical notation and evaluate.
Objective 2: Use the properties of exponents.
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: m n m nx x x Product to a Power: m m mxy x y
Quotient rule: m
m nn
xx
x
Power rule: nm mnx x
Quotient to a Power: m m
m
x x
y y
Negative power:
n nx y
y x
Simplify each expression. Assume that x is a positive real number.
20. 7
6 275
Simplify each expression. Assume that x is a positive real number.
21.31
8 849 49
Simplify each expression. Assume that x is a positive real number.
22.1 1
3 34 2
Simplify each expression. Assume that x is a positive real number.
23.13
5
85
3
3
Simplify each expression. Assume that x is a positive real number.
24.3
8 416
81
x
Simplify each expression. Assume that x and y are positive real numbers.
25.
29 327
8
x
Simplify each expression. Assume that x and y are positive real numbers.
26. 9
1 2 29 99 9
Simplify each expression. Assume that x and y are positive real numbers.
27.
13
15
x
x
Simplify each expression. Assume that x and y are positive real numbers.
28.2 1
5 2x x
Simplify each expression. Assume that x and y are positive real numbers.
29.
62 7
3
53
x
x
Simplify each expression. Assume that x and y are positive real numbers.
30. 2
9 3 34 2125x y
Simplify each expression. Assume that x and y are positive real numbers.
31. 1 1
3 25 5
22 5
xy x y
x y
Simplify each expression.
32. 3 7 325 5 5 52 9 5x x x x
Simplify each expression.
33. 1 1 1 12 2 2 23 5 3 5x y x y
Simplify each expression.
34. 2123x x
Simplify each expression.
35. 1 2 13 3 33 3 9a a a
Use a graphing calculator or a spreadsheet to approximate each expression to the nearest hundredth.