Rational Exponents
In other words, exponents that are
fractions.
Definition ofFor any real number b and any integer n > 1,
except when b < 0 and n is even
b1
n
b1
n = bn
Examples:
= 36 1. 361
2 =6
= 643 2. 64
1
3 =4
=
149
⎛
⎝ ⎜
⎞
⎠ ⎟
1
2
3. 49−
1
2
=
149
= 83
4.
18
⎛
⎝ ⎜
⎞
⎠ ⎟
−1
3
=2
Examples:
=
17
= 8( )
1
3
Economists refer to inflation as increases in the average cost of purchases. The formula C = c(1 + r)n can be used to predict the cost of consumer items at some projected
time. In this formula C represents the projected cost of the item at the given
annual inflation rate, c the present cost of the item and r is the rate of inflation (in decimal form), and n is the number of
years for the projection. Suppose a gallon of milk costs $2.69 now. How much would
the price increase in 6 months with an inflation rate of 5.3%?
Step 1: Identify the known values
Formula C = c(1 + r)n
c = $2.69 present cost of the item
r = 0.053 rate of inflation (in decimal
form)n = 1/2
# of years for the projection
Step 2: Find the value for C
Formula C = c(1 + r)n
C = 2.69(1 + 0.053)1/2
C = 2.69(1.053)1/2
C = $2.76
Answer the question How much would the price increase?
$2.76-$2.69 = $0.07 or 7¢
Definition of Rational Exponents
For any nonzero number b and any integers m and n
with n > 1,
except when b < 0 and n is even
b
m
n = bmn = bn( )
m
NOTE: There are 3 different ways to write a rational exponent
27
4
3 = 2743 = 273( )
4
2. 274
3
Examples:
= 273
( )4
1. 363
2
= 3( )
4 =81
= 36( )
3
= 6( )
3 =216
3. 813
4 = 814
( )3
= 3( )
3 =27
Simplifying Expressions No negative exponents No fractional exponents in the
denominator No complex fractions (fraction
within a fraction) The index of any remaining
radical is the least possible number
Examples: Simplify each expression
=42
6 ⋅a3
6 ⋅b5
6 41
3 ⋅a1
2 ⋅b5
6
= 426 ⋅ a36 ⋅ b56
= 16a3b56Get a common denominator - this is going to be our index
Rewrite as a radical
Examples: Simplify each expression
=x1
2+
3
4+
1
5
=x10
20+
15
20+
4
20 =x29
20
x1
2 ⋅x3
4 ⋅x1
5
=x20
20 ⋅x9
20 =x x920
Remember we add exponents
=w
1
5
w5
5
=w
1
5
w
Examples: Simplify each expression
w−4
5
=
1w
⎛
⎝ ⎜
⎞
⎠ ⎟
4
5
=1
4
5
w4
5
=1
w4
5
⋅w
1
5
w1
5
To rationalize the denominator
we want an integer exponent
=w
1
5
w4
5+
1
5 =
w5
w
=
xy7
8
y
Examples: Simplify each expression
=x
1y
⎛
⎝ ⎜
⎞
⎠ ⎟
1
8
=x⋅1
y1
8
=x
y1
8
⋅y
7
8
y7
8
To rationalize the denominator
we want an integer exponent
=x y78
y
xy−1
8
=21
2−
1
4 =22
4−
1
4 =21
4
Examples: Simplify each expression
=32
1
10
41
8
=25
( )1
10
22( )
1
8
=2
5
10
22
8
=2
1
2
21
4
3210
48
= 24
Examples: Simplify each expression
=5
−1
2
2⋅51
2
5−1
2
2 5
=12
⋅5
−1
2
51
2
=
12
⋅5−
1
2−
1
2
=
12
⋅5−1
=
12
⋅15
=110
Examples: Simplify each expression
⋅m
1
2 +1
m1
2 +1
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
=
m+1m−1
1
m1
2 −1
=
m1
2 +1m−1
Multiply by conjugate and use
FOIL
Examples: Simplify each expression
62
5 y8
5 =6
2
5 y5
5 y3
5 =6
2
5 ⋅ y⋅ y3
5
= y⋅ 625 ⋅ y35 = y 62 y35
= y 36y35
Examples: Simplify each expression
c2
3 −c
c−1
2
=c2
3 ⋅c1
2 −c⋅c1
2 =c7
6 −c3
2
=c
1
2 c2
3 −c⎛
⎝ ⎜
⎞
⎠ ⎟
Examples: Simplify each expression
=c6
6c1
6 −c2
2c1
2
=c c6 −c c
=c7
6 −c3
2
Examples: Simplify each expression
=2x
3
2
x2
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
−2
2x−2
x−3
2
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−2
= 2x
3
2−2⎛
⎝ ⎜
⎞
⎠ ⎟
−2
= 2x
−1
2⎛
⎝ ⎜
⎞
⎠ ⎟
−2
=2−2x−1
2⋅−2
=
122 ⋅x
=
x4
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