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Section 3: Fancy Exponents
Negative Exponents
When a negative number appears in the exponent of an expression, it means the reciprocal
of the expression.
Example 1:
3−2 =
1
1
=
2
3
9
5−1 =
1
5
−2
1
= 42 = 16
4
1
. an
Notice that the negative in the exponent did not change the sign on the expression. We can
use the rules that we learned in section 2 together with this idea of negative exponents to
simplify algebraic fractions.
In general, a−n =
Example 2: Simplify the following
x5
1
= x5 · 2 = x5x−2 = x5+(−2) = x5−2 = x3
2
x
x
1
y3
3
=
y
·
= y 3y −7 = y 3+(−7) = y 3−7 = y −4
7
7
y
y
In general,
xn
= xn−m
xm
Example 3: Write the following in simplest terms using only positive exponents
2
1 1
1
1
−2 2
(x ) =
= 2· 2 = 4
2
x
x x
x
2
1
1
x2
1
1
2 −2 2
2
2
(xy ) x =
x
=
·
·
x
=
=
xy 2
xy 2 xy 2
x2 y 4
y4
Fractional Exponents
The rules for handling fractional exponents are the same as the rules for handling integer
exponents.
Example 4: Simplify the following
1
1
1
1
5
x2 x3 = x2 +3 = x6
1 13
1 1
1
x2
= x2·3 = x6
1
x2
x
1
3
1
1
1
= x2 −3 = x6 . A fractional exponent is another way to represent a root.
Example 5:
√
x
√
1
42 = 4 = 2
√
1
x3 = 3 x
√
1
83 = 3 8 = 2
√
1
(81) 4 = 4 81 = 3 1
x2 =
When taking roots of expressions, always keep in mind that they are the reverse of raising
expressions to powers. In example 5 we compute the 4th root of 81 by finding the number
which, when you raise it to the 4th power, gives you 81. That is, 3 to the 4th power is 81.
Example 6: Simplify the following
√
3
8x3 = 2x because (2x)3 = 8x3
r
y 4 y 4
4
y
4 y
= because
=
16
2
2
16
You will often need to work with expressions that involve roots, so it is important for you to
know when you can simplify the expression and when you cannot. It is always legal to write
the root of a product as the product of roots. That is, you can split a root at a multiplication.
Example 7: Simplify the following
√
√ √
√
9x = 9 x = 3 x
√
√
√
√
√
3
3
3
8x4 = 8x3 · x = 8x3 3 x = 2x 3 x
√
√
4 + x2 cannot be simplified! It is tempting to think 4 + x2 = 2 + x, but if we square
2 + x, we do not get 4 + x2 . Sometimes you will want to eliminate the radicals in an expression altogether and write an
expression using fractional exponents.
Example 8: Write the following in the form xn .
1
1
√
= x− 3
3
x
√
3
x
1
2
= x 3 −1 = x− 3
x
This exponent has a lot of information: the negative means that the reciprocal;
the 2 in the numerator means square x; the 3 in the denominator means take the
1
cube root. So if x = 8, the value of this expression is . 4
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