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Section 9.5
Rational Exponents and Radicals
9.5 Lecture Guide: Rational Exponents and Radicals
Objective: Interpret and use rational exponents.
Principle nth Root
The principal nth root of the real number x is denoted by either 1nx or .n x
Verbally
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
Examples in Exponential Notation
16 1
216
3 27 1
327
0 01
20 0
3 0 01
30 0
3 64 1
364 5 32
1532
The principal nth root of the real number x is denoted by either 1nx or .n x
Verbally Examples inRadical Notation
Examples in Exponential Notation
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
Principle nth Root
Use the product rule for exponents to simplify the following expressions:
1.1 1
2 2x x
12x is the principal _________________________
_________________________ 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 _________________________
_________________________ 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 _________________________
_________________________ of x.
Write each expression in radical notation and evaluate.
4.1
216
5.1
38
Write each expression in radical notation and evaluate.
6.1
416
Write each expression in radical notation and evaluate.
7.1
249 8. 1
249
Write each expression in radical notation and evaluate.
9. 1
3125
Write each expression in radical notation and evaluate.
Rational ExponentsFor a real number x and rational numbers m and n :
Algebraically Examples inRadical Notation
Examples in Exponential Notation
If 1nx
is a real number*
1 m mmnn nx x x
or
1m nm mnnx x x
2233
2
8 8
2
4
2
23 377
67
y y
y
22 13 3
2
8 8
2
4
3 22
3 1 77
6
7
y y
y
Rational ExponentsFor a real number x and rational numbers m and n :
Algebraically Examples inRadical Notation
Examples in Exponential Notation
If 1nx
is a real number*
1mn
mn
xx
34
34
3
116
16
1
21
8
34
34
314
3
116
161
16
1
21
8
*If 0x and n is even, then 1nx is not a real number.
Write each expression in radical notation and evaluate.
10.3
216
11.2
38
Write each expression in radical notation and evaluate.
12. 2
38
Write each expression in radical notation and evaluate.
13.2
327
Write each expression in radical notation and evaluate.
14. 3
481
Write each expression in radical notation and evaluate.
15. 1
3125
Write each expression in radical notation and evaluate.
Use a graphing calculator to approximate each expression to the nearest hundredth.
16. 5 23
Use a graphing calculator to approximate each expression to the nearest hundredth.
17. 5 23
Use a graphing calculator to approximate each expression to the nearest hundredth.
18. 6 23
Use a graphing calculator to approximate each expression to the nearest hundredth.
19. 6 23
Exponent RulesYou should be familiar with the properties of integer exponents. Note that these properties apply to all real exponents, including rational exponents.Let m, n, x, y, xm, xn, ym, and yn be 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
with 0x and 0.y
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.
23.3
8 416
81
x
Simplify each expression. Assume that x and y are positive real numbers.
24.
29 327
8
x
Simplify each expression. Assume that x and y are positive real numbers.
25.13
5
85
3
3
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.
27.1
3
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
