This document discusses rational exponents. It reviews negative exponents and introduces rational exponents. Rational exponents are defined as the nth root of the base raised to the power of the numerator over the denominator. Examples are provided to evaluate expressions with rational exponents. Rules for simplifying, adding, and factoring terms with rational exponents are described.

1. 0.6 Rational Exponents
Chapter 0 Review of Basic Concepts

2. Concepts & Objectives
⚫ Rational Exponents
⚫ Review negative exponents and the quotient rule
⚫ Review rational exponents and simplify expressions
⚫ Factor expressions with negative or fractional
exponents

3. Negative Exponents
⚫ If you’ll recall from our review of exponent rules, when
we divide an expression with exponents, we subtract
exponents of the same base.
⚫ This process leads us to the idea that a negative
exponent is the inverse or reciprocal of the expression.
⚫ Examples: Evaluate each expression.
a) b) c)2
4− 2
4−
−
3
2
5
−
 
 
 

4. Negative Exponents
⚫ If you’ll recall from our review of exponent rules, when
we divide an expression with exponents, we subtract
exponents of the same base.
⚫ This process leads us to the idea that a negative
exponent is the inverse or reciprocal of the expression.
⚫ Examples: Evaluate each expression.
a) b) c)2
4− 2
4−
−
3
2
5
−
 
 
 
2
1
4
=
1
16
=

5. Negative Exponents
⚫ If you’ll recall from our review of exponent rules, when
we divide an expression with exponents, we subtract
exponents of the same base.
⚫ This process leads us to the idea that a negative
exponent is the inverse or reciprocal of the expression.
⚫ Examples: Evaluate each expression.
a) b) c)2
4− 2
4−
−
3
2
5
−
 
 
 
2
1
4
=
1
16
= 2
1
4
= −
1
16
= −

6. Negative Exponents
⚫ If you’ll recall from our review of exponent rules, when
we divide an expression with exponents, we subtract
exponents of the same base.
⚫ This process leads us to the idea that a negative
exponent is the inverse or reciprocal of the expression.
⚫ Examples: Evaluate each expression.
a) b) c)2
4− 2
4−
−
3
2
5
−
 
 
 
2
1
4
=
1
16
= 2
1
4
= −
1
16
= − 3
5
2
 
=  
 
125
8
=

7. Negative Exponents (cont.)
Examples: Simplify each expression and write answers
without negative exponents. Assume all variables
represent nonzero real numbers.
a) b)
3 1
2
12
8
p q
p q
−
−
( ) ( )
( )
1 2
2 5
2
1 2
3 3
3
x x
x
− −
− −

8. Negative Exponents (cont.)
Examples: Simplify each expression and write answers
without negative exponents. Assume all variables
represent nonzero real numbers.
a) b)
3 1
2
12
8
p q
p q
−
−
( ) ( )
( )
1 2
2 5
2
1 2
3 3
3
x x
x
− −
− −
( )3 2 1 112
8
p q
− − − −
=

9. Negative Exponents (cont.)
Examples: Simplify each expression and write answers
without negative exponents. Assume all variables
represent nonzero real numbers.
a) b)
3 1
2
12
8
p q
p q
−
−
( ) ( )
( )
1 2
2 5
2
1 2
3 3
3
x x
x
− −
− −
( )3 2 1 112
8
p q
− − − −
=
5 23
2
p q−
=
5
2
3
2
p
q
=

10. Negative Exponents (cont.)
Examples: Simplify each expression and write answers
without negative exponents. Assume all variables
represent nonzero real numbers.
a) b)
3 1
2
12
8
p q
p q
−
−
( ) ( )
( )
1 2
2 5
2
1 2
3 3
3
x x
x
− −
− −
( )3 2 1 112
8
p q
− − − −
=
5 23
2
p q−
=
5
2
3
2
p
q
=
( )( )1 2 2 10
2 4
3 3
3
x x
x
− − − −
− −
=
3 12
2 4
3
3
x
x
− −
− −
=

11. Negative Exponents (cont.)
Examples: Simplify each expression and write answers
without negative exponents. Assume all variables
represent nonzero real numbers.
a) b)
3 1
2
12
8
p q
p q
−
−
( ) ( )
( )
1 2
2 5
2
1 2
3 3
3
x x
x
− −
− −
( )3 2 1 112
8
p q
− − − −
=
5 23
2
p q−
=
5
2
3
2
p
q
=
( )( )1 2 2 10
2 4
3 3
3
x x
x
− − − −
− −
=
3 12
2 4
3
3
x
x
− −
− −
=
( ) ( )3 2 12 4
3 x
− − − − − −
= 1 8
3 x− −
=

12. Negative Exponents (cont.)
Examples: Simplify each expression and write answers
without negative exponents. Assume all variables
represent nonzero real numbers.
a) b)
3 1
2
12
8
p q
p q
−
−
( ) ( )
( )
1 2
2 5
2
1 2
3 3
3
x x
x
− −
− −
( )3 2 1 112
8
p q
− − − −
=
5 23
2
p q−
=
5
2
3
2
p
q
=
( )( )1 2 2 10
2 4
3 3
3
x x
x
− − − −
− −
=
3 12
2 4
3
3
x
x
− −
− −
=
( ) ( )3 2 12 4
3 x
− − − − − −
= 1 8
3 x− −
=
8
1
3x
=

13. Rational Exponents
⚫ Using the power rules, we can see that (for n ≠ 0)
⚫ The definition of an can thus be extended to rational
values of n (fractions) by defining a1/n to be the nth root
of a, or the number whose nth power is a.
( ) ( )1/1/ 1
n n nn
a a a a= = =

14. Rational Exponents (cont.)
⚫ For a1/n, where n is a positive integer,
⚫ Remember order of operations!
a1/n, n Even If n is even, and if a > 0, then a1/n
is the principal nth root of a.
a1/n, n Odd, If n is odd, and a is any nonzero
real number, then a1/n is the
positive or negative nth root of a.
( )
11
22
100 100−  −

15. Rational Exponents (cont.)
Examples: Evaluate each expression.
a) 361/2 b) –1251/3 c) (–625)1/4
d) –2251/2 e) 321/5

16. Rational Exponents (cont.)
Examples: Evaluate each expression.
a) 361/2 b) –1251/3 c) (–625)1/4
d) –2251/2 e) 321/5
6= 5= −
not a real number

17. Rational Exponents (cont.)
Examples: Evaluate each expression.
a) 361/2 b) –1251/3 c) (–625)1/4
d) –2251/2 e) 321/5
6= 5= −
not a real number
15= − 2=

18. Rational Exponents (cont.)
⚫ What about rational exponents where the numerator is
not 1?
⚫ The notation am/n must be defined so that all of the
previous rules for exponents still hold. For the power
rule to hold, (a1/n)m must equal am/n. Therefore, am/n
is defined as follows:
For all integers m, all positive integers n, and all
real numbers a for which a1/n is a real number,
( ) ( )
11
or
mm m
m nn n n
a a a a= =

19. Rational Exponents (cont.)
⚫ There are two ways you can evaluate an am/n expression:
⚫ Mentally: am/n means the nth root of the mth power,
so 323/5 would mean the 5th root of 32, which is 2,
raised to the 3rd power: 23 = 8.
⚫ Calculator: Most calculators have either a ^ or a xy
button. Unless you are using something like Desmos
that formats it for you correctly, make sure you put
parentheses around the fraction. See next slide for
examples.

20. Rational Exponents and Calculators
⚫ Without parentheses ⚫ With parentheses
 ✓

21. Rational Exponents (cont.)
Examples: Evaluate each expression.
a) 1252/3
b) (–64)2/3
c) –813/2
d) (–4)5/2

22. Rational Exponents (cont.)
Examples: Evaluate each expression.
a) 1252/3
b) (–64)2/3
c) –813/2
d) (–4)5/2
2
5 25= =
( )
2
4 16= − =
3
9 729= − = −
( )
1/2
not a real number because 4 is not real−

23. Simplifying Rational Exponents
Examples: Simplify.
a)
b)
c)
2/3 1/2
6 2y y
( )2/3 7/3 1/3
2m m m+
2 2/35/6 3
3/4 6
3 8v y
y v
   
   
   
2 1
3 2
12y
+
=

24. Simplifying Rational Exponents
Examples: Simplify.
a)
b)
c)
2/3 1/2
6 2y y
( )2/3 7/3 1/3
2m m m+
2 2/35/6 3
3/4 6
3 8v y
y v
   
   
   
2 1
3 2
12y
+
=
4 3
7/66 6
12 12y y
+
= =

25. Simplifying Rational Exponents
Examples: Simplify.
a)
b)
c)
2/3 1/2
6 2y y
( )2/3 7/3 1/3
2m m m+
2 2/35/6 3
3/4 6
3 8v y
y v
   
   
   
2 1
3 2
12y
+
=
4 3
7/66 6
12 12y y
+
= =
2 7 2 1
3 3 3 3
2m m
+ +
= +

26. Simplifying Rational Exponents
Examples: Simplify.
a)
b)
c)
2/3 1/2
6 2y y
( )2/3 7/3 1/3
2m m m+
2 2/35/6 3
3/4 6
3 8v y
y v
   
   
   
2 1
3 2
12y
+
=
4 3
7/66 6
12 12y y
+
= =
2 7 2 1
3 3 3 3
2m m
+ +
= +
9 3
33 3
2 2m m m m= + = +

27. Simplifying Rational Exponents
Examples: Simplify.
a)
b)
c)
2/3 1/2
6 2y y
( )2/3 7/3 1/3
2m m m+
2 2/35/6 3
3/4 6
3 8v y
y v
   
   
   
2 1
3 2
12y
+
=
4 3
7/66 6
12 12y y
+
= =
2 7 2 1
3 3 3 3
2m m
+ +
= +
9 3
33 3
2 2m m m m= + = +
5/3 2
3/2 4
9 4v y
y v
  
=   
  
5 3
4 2
3 2
36v y
− −
=

28. Simplifying Rational Exponents
Examples: Simplify.
a)
b)
c)
2/3 1/2
6 2y y
( )2/3 7/3 1/3
2m m m+
2 2/35/6 3
3/4 6
3 8v y
y v
   
   
   
2 1
3 2
12y
+
=
4 3
7/66 6
12 12y y
+
= =
2 7 2 1
3 3 3 3
2m m
+ +
= +
9 3
33 3
2 2m m m m= + = +
5/3 2
3/2 4
9 4v y
y v
  
=   
  
5 3
4 2
3 2
36v y
− −
=
5 12 4 3
3 3 2 2
36v y
− −
=
1/2
7/3 1/2
7/3
36
36
y
v y
v
−
= =

29. Factoring Rational Exponents
⚫ When factoring expressions with negative or rational
exponents, just as before, factor out the least power of
the variable or variable expression.
⚫ Example:
⚫
⚫
2 3
12 8x x− −
−
1/2 3/2
4 3m m+

30. Factoring Rational Exponents
⚫ When factoring expressions with negative or rational
exponents, just as before, factor out the least power of
the variable or variable expression.
⚫ Example:
⚫
⚫
2 3
12 8x x− −
−
1/2 3/2
4 3m m+
( ) ( )
( )2 3 3 33
4 3 2x x x
− − − − − −−
= − ( )3
4 3 2x x−
= −

31. Factoring Rational Exponents
⚫ When factoring expressions with negative or rational
exponents, just as before, factor out the least power of
the variable or variable expression.
⚫ Examples:
⚫
⚫
2 3
12 8x x− −
−
1/2 3/2
4 3m m+
( ) ( )
( )2 3 3 33
4 3 2x x x
− − − − − −−
= − ( )3
4 3 2x x−
= −
( )1/2 1/2 1/2 3/2 1/2
4 3m m m− −
= + ( )1/2
4 3m m= +

32. Classwork
⚫ 0.6 Assignment - Pg. 59: 4-36 (even); pg. 51: 40-58
(even); pg. 42: 80-104 (×4)
⚫ 0.6 Classwork Check (due 9/14)
⚫ Quiz 0.5 (due 9/14)