Download Fractional Exponents and Radicals

Fractional Exponents and
Radicals
Fractional Exponents:
Calculator Warning
Calculate the following in your calculator:
 1 2
2
 1.41421
Not Exact. What is the
1/2?
exact
value
of
2
2^(1÷2)
½
A Math Trick to Evaluate 2 Exactly
Evaluate the following using Exponential Laws:
1/2.
1/2
2 2
=
1/2
+
1/2
2
When
multiplying, if the
bases are the
same add the
exponents
1
2
2
Now use this result to
figure out what
21/2 equals.
½
What Does 2 Exactly Equal?
Since:
2 2  2
2
12
2   2
12
12
2 
2
12
12
2
The final answer makes
sense because:
 2
 2
2 2 2
What is the number
whose square is 2:
x2 = 2 ?
Irrational Numbers
Radical numbers/Fractional Roots are typically
irrational numbers (unless they simplify to an
integer). Our calculator gives:
2 2
1/ 2

1.41421
But the decimal will go on forever because it is an
irrational number. For the exact answer just use:
2
More Fractional Exponents
Change the following into a radical:
1/3
2
Remember 21/2
is also called
the square
root of 2
=
3
2
The cube root of 2.
3
The solution to x = 2
nth Roots
The nth root of a number x is a number r
which, when multiplied by itself n times
equals x.
n
x  r if r  x
n
Examples:
3
125  5 since 5  5  5  5  125
3
5
32  2
since 2  2  2  2  2  2  32
5
Radical v. Exponential Notation
Rewrite the following roots as an exponential expression:
Fourth root
4
Fifth root
5
Tenth root
10
Seventeenth root
17
nth root
n
x x
1
x x
1
x x
4
5
1
10
x x
1
17
x x
1
n
Square Root Notation
Remember:
2
x x
The 2 is implied!
Roots in the Calculator
16
12
 12
6
Exponential Notation:
1
2
^
(
1
÷
6
)
(
1
2
)
Radical Notation:
(
x
6
)
x
is found in MATH #5
Fractional Exponents with Numerators
greater than 1
Approximate the following expression:
5/3
2
≈ 3.174502
Find the exact value:
2
5
3
1
Expand
1
1
1
 2 2 2 2 2
3
3
3
3
1
3
 2 2 2 2 2
3

3
 2
3
3
5
Simplify
3
3
Rewrite in Radical
Notation
Exponential v. Radical Notation
b
 b
q
p/q
p
or
=
q
b 
p
Generally b≥0
Example
Evaluate the following without a
calculator:
Rewrite in Radical
Notation
5 6
64

6
Simplify
   2
5
64
5
 32
By rewriting the fractional exponential expression into a
radical expression, the arithmetic is more
straightforward.