This module covers the laws the govern the manipulation of exponents in algebra.
The following are generally referred to as the “laws” or “rules” of exponents.
As with any formula, the most important thing is to be able to
use them—that is, to understand what they mean. But it is also important to know
where these formulae come from . And finally, in this case, the three should be memorized.
So...what do they mean? They are, of course, algebraic generalizations—statements that are true for any
,
, and
values. For instance, the first rule tells us that:
which you can confirm on your calculator. Similarly, the third rule promises us that
These rules can be used to combine and simplify expressions.
Simplifying with the Rules of Exponents |
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Third rule :
,
where
and
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First rul e:
,
done on both the top and bottom |
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Second rule :
,
where we choose this form to avoid a negative exponent |
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Why do these rules work? It’s very easy to see, based on what an exponent is.
Why does the first rule work? |
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You see what happened?
means three 19s multiplied;
means four 19s multiplied. Multiply them together, and you get
seven 19s multiplied.
Why does the second rule work? |
First form |
Second form |
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In this case, the key is
fraction cancellations . When the top is multiplied by 19 and the bottom is multiplied by 19, canceling these 19s has the effect of dividing
the top and bottom by 19. When you divide the top and bottom of a fraction by the same number, the fraction is unchanged.
You can also think of this rule as the inevitable consequence of the
first rule. If
, then
(which asks the question “
times what equals
?”) must be
.
Why does the third rule work? |
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What does it mean to raise something to the fourth power? It means to multiply it by itself, four times. In this case, what we are multiplying by itself four times is
, or
. Three 19s multiplied
four times makes twelve 19s multiplied.