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# Exponent properties 3

Video transcript

Simplify x to the
third, and then that raised to the fourth
power times x squared, and then that raised
to the fifth power. Now, here we're going to use
the power property of exponents, sometimes called
the "power rule." And that just tells us
if I have x to the a, and then I raise
that to b-th power, this is the same thing as
x to the a times b power. And to see why that works,
let's try that with a-- well, I won't do it with
these right here. I'll do it with a
simpler example. Let's say we are
taking x squared and then raising that
to the third power. Well, in that situation,
that literally means multiplying x squared
by itself three times. So that literally
means, x squared times x squared times x squared. I'm taking x squared and I'm
raising it to the third power. Now, what does this
mean over here? Well, there's a couple
of ways to do it. You could just say, look,
I have the same base, I'm taking the product, I
can just add the exponents. So this is going to be
equal to-- let me do it in that same magenta--
this is equal to x to the 2 plus 2 plus 2 power, or
essentially x to the 3 times 2. This right here
is just 3 times 2. So we get x to the sixth power. And if you want,
you say, hey, Sal, I don't understand why you
can add those exponents. You just have to remember that x
squared-- this thing right over here, we could
rewrite as x times x times x times x--
in parentheses, I'm putting each of these x
squareds-- times x times x. And this is just x
times itself 6 times x times x times x times
x times x times x. This is just x to
the sixth power. So that's why we can add
the exponents like that. So let's just use
that power property on this expression
right over here. We start off with we have
x to the third raised to the fourth power. So that's just going to be x
to the 3 times 4 power, or x to the 12th power. Then, we're
multiplying that by x squared raised to
the fifth power. Well, that's just going to be
x to the 2 times 5 power, or x to the 10th power. And now we have the same base,
and we're taking the product, we can just add the exponents. This is going to be equal
to-- this whole expression is going to be
equal to x to the 12 plus 10th power, or
x to the 22nd power. And we are done.