Download Negative & Rational Exponents

Rational Exponents
Now let’s consider rational exponents.
Consider the following example:
64  3 64  4
1
3
These are different forms of the same thing.
They both mean “the third root of 64”. That is,
what number, when multiplied by itself 3 times,
equals 64. In this case the answer is 4
In general, this can be written in the following way:
This is the
exponential form
a  a
1
b
b
This is the
radical form
This is considered the “bth root of a”. In other words, what
number, when multiplied by itself “b” times, equals “a”.
Rational Exponents
Write as a radical then evaluate the following:
All scientific
calculators have an
nth root button. It is
usually a 2nd function
key. Sometimes it
looks like 1 y or x y
729 
1
6
6
729  3
1331  4 1331  11
1
4
Roots are
not always
whole
numbers
36  36  2.05
1
1
1

4
  0.5
16 
16 2
16
1
5
x
The way you type it
in will vary from
calculator to
calculator so check
your manual.
5
1
4
1
4
1
NOTE: when we have x 2 this is
a square root and we typically
write x instead of 2 x
The negative sign could pose a problem
for some. Be sure to write with positive
exponents first, then as a root.
Rational Exponents
When the rational expression is a little more complex,
we alter our form slightly:
5
4
This is now rewritten by separating the fraction into
two parts, (¼)x5. Thus becoming a power of a power:
81
81  81
5
4
 
1
4
5
Now we can do the root inside the
brackets normally and then put the
answer to the exponent 5.
81  3  243
5
4
5
 
81  81
5
4
5
1
4
This could have been done
in the reverse order too:
 (3486784401)  243
1
4
Although the above line is not wrong, it is typically not done this
way so that the roots don’t have to be so big.
Rational Exponents
Try these examples without using a calculator:

 
625 
3
4
7
10
1024
3
12
729x

4
625  5  125
10
1024  2  128
3
3

7

 729x
12
7
   729x 
1
3
1
2
 729 x  3x
1
6
12
6
2
12
2
 125   125    5   25
 

  3

36
 216   216   6 
2
3
3
Try the
questions first,
don’t just click.
2
1
6
This is not the
only order to do
this one.