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# Exponential equation with rational answer

Video transcript

So right here, we've got 3 to
the a power-- or a-th power, I guess. I don't want to confuse
it with the number 8. 3 to the a power is equal to
the fifth root of 3 squared. And what we need to figure out
is what would a be equal to. Let's solve for a. And I encourage you right
now to pause this video and try it on your own. Well, if you have a fifth
root right over here, one thing that you
might be tempted to do to undo the fifth root is
to raise it to the fifth power. And of course, we can't just
raise one side of an equation to the fifth power. Whatever we do to
one side, we have to do the other side if we
want this to still be equal. So let's raise both
sides of this equation to the fifth power. Now, this left-hand
side-- we just have to remember a little bit
of our exponent properties. 3a to the fifth power. And if we want to
just remind ourselves where that comes from,
that's the same thing as 3 to the a times 3 to the a
times 3 to the a times 3 to the a times 3 to the a. Well, what's that
going to be equal to? That's going to be 3 to
the a plus a plus a plus a plus a power, which is the same
thing as 3 to the 5a power. So the exponent
property here is if you raise a base to some
exponent and then raise that whole thing to
another exponent, that's the equivalent of raising
the base to an exponent that is the product of
these two exponents. So we can rewrite this left-hand
side as 3 to the 5a power is going to be equal to-- Well,
if you take something that's a fifth root and you raise
it to the fifth power, then you're just left with
what you had under the radical. That's going to be
equal to 3 squared. So now things become
a lot clearer. 3 to the 5a needs to
be equal to 3 squared. Or another way of
thinking about it-- we have the same
base on both sides. So this exponent needs to be
equal to this exponent right over there. Or we could write that 5 times
a needs to be equal to 2. And of course, now we can
just divide both sides by 5. And we get a is equal to 2/5. And this is an
interesting result. And what's neat
about this example-- it kind of shows
you the motivation for how we define
rational exponents. So let's just put this back
into the original expression. We've just solved for a. And we've gotten that 3 to
the 2/5 power-- and actually, let me color code
it a little bit because I think
that'll be interesting. 3 to the 2/5 power is
equal to the fifth root-- notice, the fifth root. So the denominator
here, that's the root. So the fifth root of 3 squared. So if you take this
base 3, you square it, but then you take the
fifth root of that, that's the same thing as
raising it to the 2/5 power. Notice. Take this 3, take it
to the second power, and then you find
the fifth root of it. Or if you use this property that
we just saw right over here, you could rewrite this. This is the same
thing as 3 squared. And then you raise
that to the 1/5 power. We saw that property
at play over here. You could just multiply
these two exponents. You'd get 3 to the 2/5 power. And that's the same
thing as 3 squared and then find the
fifth root of it. 3 squared, and then
you're essentially finding the fifth root of it.