2.  Exponents are shorthand for repeated
multiplication of the same thing by itself. For
instance, the shorthand for multiplying three
copies of the number 5 is shown on the right-
hand side of the "equals" sign in (5)(5)(5) = 53.
The "exponent", being 3 in this example,
stands for however many times the value is
being multiplied.The thing that's being
multiplied, being 5 in this example, is called
the "base".

3.  This process of using exponents is called
"raising to a power", where the exponent is
the "power".The expression "53" is
pronounced as "five, raised to the third
power" or "five to the third".There are two
specially-named powers: "to the second
power" is generally pronounced as "squared",
and "to the third power" is generally
pronounced as "cubed". So "53" is commonly
pronounced as "five cubed".

4.  When we deal with numbers, we usually just
simplify; we'd rather deal with "27" than with
"33". But with variables, we need the
exponents, because we'd rather deal with "x6"
than with "xxxxxx".
 Exponents have a few rules that we can use
for simplifying expressions.
 Simplify (x3)(x4) Copyright © Elizabeth
Stapel 2000-2011 All Rights

5.  To simplify this, I can think in terms of what
those exponents mean. "To the third" means
"multiplying three copies" and "to the fourth"
means "multiplying four copies". Using this
fact, I can "expand" the two factors, and then
work backwards to the simplified form:
 (x3)(x4) = (xxx)(xxxx)
= xxxxxxx
= x7

6.  Note that x7 also equals x(3+4).This
demonstrates the first basic exponent rule:
Whenever you multiply two terms with the
same base, you can add the exponents:
 ( xm ) ( xn ) = x( m + n )

7.  However, we can NOT simplify (x4)(y3), because
the bases are different: (x4)(y3) = xxxxyyy =
(x4)(y3). Nothing combines.
 Simplify (x2)4
 Just as with the previous exercise, I can think in
terms of what the exponents mean.The "to the
fourth" means that I'm multiplying four copies
of x2:
 (x2)4 = (x2)(x2)(x2)(x2)
= (xx)(xx)(xx)(xx)
= xxxxxxxx
= x8

8.  Note that x8 also equals x( 2×4 ).This
demonstrates the second exponent rule:
Whenever you have an exponent expression
that is raised to a power, you can multiply the
exponent and power:
 ( xm )n = xm n

9.  If you have a product inside parentheses, and
a power on the parentheses, then the power
goes on each element inside. For
instance, (xy2)3 = (xy2)(xy2)(xy2) = (xxx)(y2y2y2)
= (xxx)(yyyyyy) = x3y6 = (x)3(y2)3. Another
example would be:

10.  Warning:This rule does NOT work if you have a sum
or difference within the parentheses. Exponents,
unlike mulitiplication, do NOT "distribute" over
addition.
 For instance, given (3 + 4)2, do NOT succumb to the
temptation to say "This equals 32 + 42 = 9 + 16 = 25",
because this is wrong. Actually, (3 + 4)2 = (7)2 = 49,
not 25.When in doubt, write out the expression
according to the definition of the power. Given (x –
2)2, don't try to do this in your head. Instead, write it
out: "squared" means "times itself", so (x – 2)2 = (x –
2)(x – 2) = xx – 2x – 2x + 4 = x2 – 4x + 4.

11.  The mistake of erroneously trying to
"distribute" the exponent is most often made
when the student is trying to do everything in
his head, instead of showing his work. Do
things neatly, and you won't be as likely to
make this mistake.
 There is one other rule that may or may not
be covered at this stage:
 Anything to the power zero is just "1".

12.  though, this rule means that some exercises
may be a lot easier than they may at first
appear:
 Simplify [(3x4y7z12)5 (–5x9y3z4)2]0
 Who cares about that stuff inside the square
brackets? I don't, because the zero power on
the outside means that the value of the entire
thing is just 1.
