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Simplify 3a times a to
the fifth times a squared. So the exponent
property we can use here is if we have the same
base, in this case, it's a. If we have it raised
to the x power, we're multiplying it
by a to the y power, then this is just going to
be equal to a to the x plus y power. And we'll think about why
that works in a second. So let's just apply it here. Let's start with the a to
the fifth times a squared. So if we just apply
this property over here, this will result in a to
the fifth plus two-th power. So that's what those guys
reduce to, or simplify to. And of course, we still
have the 3a out front. Now what I want to do is
take a little bit of an aside and realize why this worked. Let's think about what a to the
fifth times a squared means. A to the fifth
literally means a times a times a times a times a. Now, a squared literally
means a times a. And we're multiplying
the two times each other. So we're multiplying these
five a's times these two a's. And what have we just done? We're multiplying a
times itself five times, and then another two times. We are multiplying
a times itself. So let me make it clear. This over here is
a to the fifth. This over here is a squared. When you multiply
the two, you're multiplying a by itself
itself seven times. 5 plus 2. So this is a to the seventh
power. a to the 5 plus 2 power. So this simplifies to 3a
times a to the seventh power. Now you might say, how do I
apply the property over here? What is the exponent on the a? And remember, if I just
have an a over here, this is equivalent to
a to the first power. So I can rewrite 3a is 3
times a to the first power. And now it maybe makes
it a little bit clearer. A to the first power-- and
the association property of multiplication, I can
do the multiplication of the a's before I
worry about the 3's. So I can multiply
these two guys first. So a to the first times
a to the seventh-- I just have to add the exponents
because I have the same base and I'm taking the
product-- that's going to be a to
the eighth power. And I still have
this 3 out front. So 3a times a to
the fifth times a squared simplifies to
3a to the eighth power.