rational exponents fraction exponents. radical expression and exponents by definition of radical...
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Rational ExponentsRational Exponents
Fraction ExponentsFraction Exponents
Radical expression and Exponents
By definition of Radical Expression.
The index of the Radical is 3.
51251255 33 so
How would we simplify this expression?
What does the fraction exponent do to the number?
The number can be written as a Radical expression, with an index of the denominator.
2
1
9
2 9
The Rule for Rational Exponents
46464 33
1
1
nn bb
Write in Radical form
2
1
6
1
m
a
Write in Radical form
mm
aa
2
1
66
1
Write each Radical using Rational Exponents
w
b5
Write each Radical using Rational Exponents
2
1
5
15
ww
bb
What about Negative exponents
Negative exponents make inverses.
7
1
49
149
2
12
1
What if the numerator is not 1Always remember:
Evaluate
5 25
2
3232
power
root
What if the numerator is not 1
Evaluate
322
5 = 3225
25( )
25 = 2105
What if the numerator is not 1
Evaluate
322
5 3225
25( )
25 2105 2
105
22 4
power
root
For any nonzero real number b, and integer m and n
Make sure the Radical express is real, no b<0 when n is even.
bm
n = bmn or bn( )
m
Simplify each expression
No fraction in the denominators.
3
2
7
4
7
1
x
yy
Simplify each expression
No fraction in the denominators.
y1
7 ⋅y47 y
57 y57
x−23
1
x23
1
x23
Simplify
64x63
Simplify
64x63
(64x6 )1
3
641
3 (x6 )1
3 =4x63 =4x2
Simplify
54x93
Simplify in simplest radical form
54x93
27 • 2 • x93
27 • 2 • x9( )
13 =27
132
13 x
93
Radical form:
54x93
27 • 2 • x93
27 • 2 • x9( )
13 =27
132
13 x
93 =3x3 23
Simplifying exponential expressions
• If in radical form, transform to exponential form
• Simplify by applying the exponent to each base.
• Express with positive exponents or radical form, depending on the directions
Express in simplest form
• Make exponential first!
25x−4
36y8
Express in simplest form
• Make exponential first!
• Now simplify
25x−4
36y8=
25x−4
36y8
⎛
⎝⎜
⎞
⎠⎟
12=
2512 x
−42
3612 y
82
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
Express in simplest form
• Make exponential first!
• Now simplify
25x−4
36y8=
25x−4
36y8
⎛
⎝⎜
⎞
⎠⎟
12=
2512 x
−42
3612 y
82
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
5x−2
6y4=
56x2y4
Next example
• Use positive exponents only!
x4y12
z−26
Next example
• Use positive exponents only!
• Or in radical form:
x4y12
z−26 =
x4y12
z−2
⎛
⎝⎜
⎞
⎠⎟
16=
x46 y
126
z−26
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟=
x23y2
z−13
=x23y2z
13
y2 x2z3
Simplify
1
1
2
1
2
1
y
y
Simplify
1
1
1
1
2
1
2
1
2
1
2
1
y
y
y
y
Simplify
1
1
1
1
1
11
2
1
2
11
2
1
2
1
2
1
2
1
y
yyy
y
y
y
y
Simplify
1
12
1
1
1
1
1
1
2
1
1
2
1
2
11
2
1
2
1
2
1
2
1
y
yy
y
yyy
y
y
y
y
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