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Exponents and Rational Exponents
What are Exponents?
4
41
4
2
43
4
4
45
3
4 4 4
4
×
×
The chart to the left shows the first five exponents of 4. In
each case, we find the exponent gives us the number of fours
found in the multiplication.
4 4 16
4 4 4 64
4 4 4 4 256
4 4 4 4 4 1024
Note that this means that 4 to the power of 1 is simply equal
to 4. In fact, any number to the exponent of 1 is equal to
itself.
And so on...
What are Rational Exponents?
4
1
2
4
because 2
2
2
4
Some rational exponents are the same as roots. The
denominator of a rational exponent (shown as a fraction) can
be thought of as a root of the same order. The example on the
left can be read as “4 to the power ½” or “the square root of 4.”
These two statements are equivalent.
The example on the left is asking what number, multiplied by
itself, would give you the base number. In this case the answer
is 2 because 2 × 2 is 4.
The Square Root
Finding the square root of a number is the opposite of squaring
a number i.e putting it to the power 2. The square root of any
number will be the number that, when squared (multiplied by
itself) gives you that number back. For example, the square
root of ten is approximately 3.1623 since when you square that
number, you get ten.
10
( 2) 2
3.1623
4 but 4
We should also note, at this time, that negative numbers can be squared as well to produce
positive numbers. For example, negative 2 squared will give you 4. It is because of this that
square roots are always defined to be positive.
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The Square Root of a Negative Number
You CANNOT find the square root of a negative number. Negative numbers do not
have square roots because there is no number that, when multiplied by itself, results in
a negative number. The square roots of negative numbers are called “imaginary
numbers”.
Roots
The cube root is just like the square
root except that now you are looking
for a number that, when cubed
(multiplied by itself twice) will give
you the original number back. See
the examples to the right. As you
may have suspected, you can also
have higher roots such as the 4th
root, 5th root etc., and they work the
same way.
3
8
2 because 2 3
3
10
2.154 because 2.1543
5
32
6
15625
2 because 2 5
8
10
32
5 because 56
15625
Multiplying and Dividing Exponents:
43
42
44 42
4
3
4
44
2
42
43
44
2
2
45
42
=43
=42
(4 4 4) (4 4) 4 5
(4 4 4 4)
4 4 42
(4 4)
Write the division
as a fraction
Some of the fours
cancel out here.
There is a special rule for multiplying and dividing
exponents with the same base. For multiplying,
you keep the base the same and add the exponents as
shown on the left. For dividing you subtract the
exponents instead of adding them.
To understand why this works, try writing out the
exponents as multiplications and see what happens.
Two examples are shown to the left.
When you count up the number of fours, you realize
that it’s the same result.
Nested Exponents:
A nested exponent occurs when you have two
exponents on the same number. In the example to
the right, you have an exponent 42 and then that
number is put to the exponent 3.
(4 2 ) 3 4 2 3 46
In this situation, the rule is that you multiply the
exponents together. So, in this example we get 46
and in the next we get 210.
(25 ) 2 25 2 210
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Deconstructing Exponents:
The last rule regarding exponents has to do with what
happens when you take an exponent of two numbers
being multiplied together (within brackets). For
example, (xy)2 = x2y2 where x and y are any two
numbers, as in the example to the right.
Another thing to keep in mind is that numbers in
exponents are sometimes perfect squares, cubes, etc.
and can be made into exponents with a smaller base
using the rule for nested exponents. Look to the right
and below for examples.
126
(3 4)6
36 (22 )6
(2 5) 2 22 52
154 (3 5) 4 34 54
163
(2 4 ) 3
212
36 212
Multiplying and Dividing with Rational Exponents
When the numerator (of the exponent) is 1:
When multiplying and dividing rational
exponents the same rules apply as with
normal exponents. Provided they have
the same base, if you are multiplying
you add the exponents together, and if
you are dividing you subtract them.
2
1
2
2
2
1
2
2
2 2
When the numerator is not 1:
To understand what you get with a fraction
such as 2/3, you must use the rules about
nested exponents and decomposing
exponents. You can see from the example
to the right how a fractional exponent can
be broken up into two exponents. From
there, you can figure out the answer.
2
Knowing:
Leads to:
1
2
21
2
By the definition of a square
root, the square root multiplied
by itself will give you the
original number.
If you are given an expression with a
square root, cube root, or any other root,
you can convert them using the pattern
described to the right.
3
1
2
2
1
3
4
(2 3 ) 2
2
(2 )
1
3
2
2
1
4
23 2
2
2
And so on….
26
1
3
2
2
3
More Examples are at the bottom of the
last page.
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Student Services Building (SSB), Room 204
905.721.2000 ext. 2491
This document last updated: 12/22/2010
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Using the Rules to Solve Expressions
Exponents:
Armed with all of these exponent rules,
you can solve a multitude of expressions
much more easily, and often without a
calculator. Just look at all of these
stunning examples!
6
11
3
2
216
121
63
113
(6 3 ) 2
11 2
5 12 25
142
72
63 6
111
5
512 (52 ) 5 512 5
(7 2) 2
72
6
63
11
7 2 22
72
1
x
Move the 2 over to
the right
2 51
x 2 49
Square root both sides
x
49
x 7, 7
Put in the plus/minus
( x 2) 2
x 2
x
Two different
solutions
36
52 25
4
1
216 11
63 11
Root Expressions:
Solving equations that result in roots can be tricky.
Consider the example to the right. You are allowed to
take the square root of both sides to solve an equation,
BUT whenever you do, you must remember to add a
plus/minus sign to one side. This means we will end up
with two different answers. In this example, we get
both 5 and -5 as solutions and both of these numbers
satisfy the original equation.
2
22
10
x2
25
x
x
25
5, 5
Start by square rooting both sides
6
Make sure to put in the plus/minus
Move the 2 over to the right side
6 2
x 6 2 and x
x 4 and x
6 2 Here, we branch into two
solutions: one with the
plus and one with the
minus.
8
This gives us two solutions, as expected.
Rational Exponent Examples:
2
2
3
4
1
2 3
(2 )
5
2
1
2
(4 )
4
5
1
3
2
3
5
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4 1.5874
32
3
0.4
3
4
10
3
2
5
3
1
2 5
9
1
5
1.1958
In this example we have a decimal exponent. This
shouldn’t cause a problem since we can just convert
the decimal (0.4) into a fraction 4/10 = 2/5.
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This document last updated: 12/22/2010