Let's look at the properties of integer exponents. For n and m integers and a and
b real numbers, we have the following properties. The first property states that
a^n a^m = a^n+m and is often referred to as the Product Rule. For example, a^2
a^3 = a^2+3 or a^5 The. second property states that a^n raised to the mth power is
equal to a^nm and is often referred to as the Power of a Power Rule. For example,
a^2 cubed is equal to a^23 or a^6. Notice that in the first property, when
the base is the same, we add the exponents. Whereas, in the second
property, when we have a power of a power, we multiply the exponent. These properties
are often confused, so know when to add and when to multiply. The third property
states that a b to the nth power is equal to a^m b^m, and is often referred
to as the Power of a Product Rule. For example, ab squared is equal to a^2
b^2. We raised both factors to the power of
two. The fourth property states that a / b
raised to the mth power, is equal to a^m / b^m, and is often referred to the Power of
a Quotient Rule. For example, a / b raised to the third power is equal to a^3 / b^3.
And here, we're assuming, of course, that b is not equal to zero. Alright, the fifth
property states, that a^m / a^n = a^m-n and is often referred to as the Quotient
Rule. For example, a^5 / a^2 = a^5-2 or a^3.
So, when the bases are the same and we're dividing, we subtract the exponents. The
sixth property states that a^0 = one, for a not equal to zero.
Zero^0 is not defined for various reasons. And this is often referred to as the Zero
Exponent Rule. For example, three^0 = one. And the last property to consider here is
that a^-n = one / a^+n, and is often referred to as the Negative Exponent Rule.
For example, a^-3 = one / a^3. Alright. Let's see an example. Let's
simplify the following expression and write our answer using only positive
exponents. Since multiplication is both commutative and associative, we can
regroup this multiplication as follows. We can take all the numbers first, the two,
the three and the five, and multiply them. So, this is equal to two three five.
And then, multiply the w terms together, so times w^-5 w^4.
And then, group the v terms together. We have this term and this term, so times
v^-6 v^7. And finally, we'll group the u terms
together. So, times u^7 u^2. So, this is equal tot two three five
is 30, and then, times w to the -five + four by the Product Rule. Because these
bases are the same, we can add those exponents. Same with the v term, so it'll
be v to the -six + seven. And finally, we'll do the same with the u
term, so it's u to the seven + two, which is equal to 30 w^-1 v^+1 u^9.
And then, by the Negative Exponent Rule, this is equal to 30 one / w^1. Remember,
we want to write our answer using only positive exponents. And when the exponent
of a variable is one, we usually do not write it. So, writing this as one fraction
and dropping those exponents of one gives us our answer of 30vu^9 / w. Alright,
let's see another example. Let's simplify this expression and write our answer using
only n positive exponents. Well, the first thing we can do is simplify what's inside
these parentheses by, again, grouping like terms. So, this is equal to, let's group
our number together, so six / three and then times, grouping our m terms together,
we have m divided by m^-1 and then finally, grouping the n terms together, we
have n^-2 / n^2, still raised to the negative third power, which is equal to
six / three is two. And then, times m^1 minus a -one.
And this comes from the Quotient Rule because the bases are the same, we
subtract the exponents. And we'll do the same with the n term. So, it's n^-2 - two,
whole thing still raised to the negative third power. And this is equal to 2m^2,
cuz it's one minus a -one times n^-4, whole thing to the negative third.
And then, by the Power of a Product Rule, we can raise each of the factors to the
negative third power, which is equal to one / two to the positive third power, by
our Negative Exponent Rule. And then, times, we have a power of a power, so
remember, we multiply two -three, which is -six.
Same with the n term, we hav e a power of a power so we multiply. So, we have -four
-three which is +twelve. And remember, we want to write our answer
using only positive exponents. So, let's use that Negative Exponent Rule again on
this m term, so this is equal to, we have one over, two^3 is eight, and then we have
one / m^6 n^12. And writing it as one fraction, will give
us are answer of n^12 / eight m^6. And this is how we work with integer
exponents. Thank you, and we'll see you next time.