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Let's look at the properties of integer
exponents. For n and m integers and a and
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b real numbers, we have the following
properties. The first property states that
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a^n a^m = a^n+m and is often referred to
as the Product Rule. For example, a^2
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a^3 = a^2+3 or a^5 The. second property
states that a^n raised to the mth power is
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equal to a^nm and is often referred to as
the Power of a Power Rule. For example,
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a^2 cubed is equal to a^2*3 or a^6.
Notice that in the first property, when*
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the base is the same, we add the
exponents. Whereas, in the second
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property, when we have a power of a power,
we multiply the exponent. These properties
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are often confused, so know when to add
and when to multiply. The third property
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states that a b to the nth power is
equal to a^m b^m, and is often referred
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to as the Power of a Product Rule. For
example, a*b squared is equal to a^2 *
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b^2.
We raised both factors to the power of
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two.
The fourth property states that a / b
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raised to the mth power, is equal to a^m /
b^m, and is often referred to the Power of
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a Quotient Rule. For example, a / b raised
to the third power is equal to a^3 / b^3.
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And here, we're assuming, of course, that
b is not equal to zero. Alright, the fifth
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property states, that a^m / a^n = a^m-n
and is often referred to as the Quotient
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Rule. For example, a^5 / a^2 = a^5-2 or
a^3.
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So, when the bases are the same and we're
dividing, we subtract the exponents. The
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sixth property states that a^0 = one, for
a not equal to zero.
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Zero^0 is not defined for various reasons.
And this is often referred to as the Zero
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Exponent Rule. For example, three^0 = one.
And the last property to consider here is
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that a^-n = one / a^+n, and is often
referred to as the Negative Exponent Rule.
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For example, a^-3 = one / a^3.
Alright. Let's see an example. Let's
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simplify the following expression and
write our answer using only positive
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exponents. Since multiplication is both
commutative and associative, we can
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regroup this multiplication as follows. We
can take all the numbers first, the two,
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the three and the five, and multiply them.
So, this is equal to two three five.
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And then, multiply the w terms together,
so times w^-5 w^4.
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And then, group the v terms together. We
have this term and this term, so times
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v^-6 v^7.
And finally, we'll group the u terms
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together. So, times u^7 u^2.
So, this is equal tot two three five
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is 30, and then, times w to the -five +
four by the Product Rule. Because these
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bases are the same, we can add those
exponents. Same with the v term, so it'll
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be v to the -six + seven.
And finally, we'll do the same with the u
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term, so it's u to the seven + two, which
is equal to 30 w^-1 v^+1 u^9.
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And then, by the Negative Exponent Rule,
this is equal to 30 one / w^1. Remember,
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we want to write our answer using only
positive exponents. And when the exponent
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of a variable is one, we usually do not
write it. So, writing this as one fraction
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and dropping those exponents of one gives
us our answer of 30vu^9 / w. Alright,
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let's see another example. Let's simplify
this expression and write our answer using
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only n positive exponents. Well, the first
thing we can do is simplify what's inside
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these parentheses by, again, grouping like
terms. So, this is equal to, let's group
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our number together, so six / three and
then times, grouping our m terms together,
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we have m divided by m^-1 and then
finally, grouping the n terms together, we
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have n^-2 / n^2, still raised to the
negative third power, which is equal to
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six / three is two.
And then, times m^1 minus a -one.
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And this comes from the Quotient Rule
because the bases are the same, we
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subtract the exponents. And we'll do the
same with the n term. So, it's n^-2 - two,
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whole thing still raised to the negative
third power. And this is equal to 2m^2,
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cuz it's one minus a -one times n^-4,
whole thing to the negative third.
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And then, by the Power of a Product Rule,
we can raise each of the factors to the
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negative third power, which is equal to
one / two to the positive third power, by
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our Negative Exponent Rule. And then,
times, we have a power of a power, so
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remember, we multiply two -three, which
is -six.
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Same with the n term, we hav e a power of
a power so we multiply. So, we have -four
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-three which is +twelve.
And remember, we want to write our answer
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using only positive exponents. So, let's
use that Negative Exponent Rule again on
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this m term, so this is equal to, we have
one over, two^3 is eight, and then we have
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one / m^6 n^12.
And writing it as one fraction, will give
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us are answer of n^12 / eight m^6.
And this is how we work with integer
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exponents. Thank you, and we'll see you
next time.