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# Exponential equation with rational answer

A worked example of rewriting a radical as an exponent. In this example, we solve for the unknown in 3ᵃ = ⁵√(3²). Created by Sal Khan.

## Want to join the conversation?

- This is still very confusing. Could someone give me a simpler explanation please?(12 votes)
- I think the video is making it much harder than it needs to be. You don't need to solve it as you would solve a typical equation (doing things to each side). You only have to consider the "definition" of fractional exponents, if that is what you would call it. This is because the base is the same for both of them (3). The denominator of the exponent will always be whichever root is taken (fifth in this case). The numerator is what the number is being raised to (2). Therefore, the exponent (a) is 2/5. Does that help?(40 votes)

- If 3^a is raised to the 5th power, shouldn't it be 243^5a and not 3a^5a?(15 votes)
- 3^a only has one term to distribute the 5th power to. You might be thinking of something like (3a)^5, which has two terms and can be expressed as (3 * a)^5, which would simplify to 243a^5.

Since 3^a is only one term, raised to the 5th power it is (3^a)(3^a)(3^a)(3^a)(3^a), or 3^5a. Hope that helps.(11 votes)

- Why didn't you just simplify the right side of the equation and discover what the exponent was? You seemed to complicate this particular problem...(15 votes)
- Yes you are correct because you still would have common bases equal to each other.(5 votes)

- I don't understand cube roots and their radicals,

²⎷3²

how do I solve this equations above?

what does the exponent behind the equation mean?(4 votes)- A cube root is asking, "What number multiplied by itself
**three**times is equal to this?". Also, your "How do I solve this equations above?" question does not apply, since, it is not an equation.(13 votes)

- Is my solution correct? ,

3ᵃ = ⁵√(3²).

3ᵃ = (3²)^(1/5).

3ᵃ = (3)^(1/5 * 2).

3ᵃ = (3)^(2/5).

a= 2/5

I only re-wrote the right-hand-side of the equation in a different form but it is still equivalent.

So I belive I don't need to change anything on the Left-hand-side since the right-hand-side still has the same value right ;)?

thanks in advance.(6 votes)- Your solution is correct. In fact, it's the same solution as alexa.pomerantz's above.(2 votes)

- couldnt you have just gone 3^2/5 instead of all that other stuff?(5 votes)
- Yes, your approach works. Remember, this is an intro video and other students may need an alternate approach to understand the relationships.(9 votes)

- This might sound like a silly question but it's worth asking.

It's very likely I'm misunderstanding something.

In a previous video sal mentioned that if you have something to a fractional exponent (i.e 5^ 1/3), that's the same thing as writing it as the cube root of 5. Or at least I think that's what he said.

In the video, I tried to figure it out on my own and thought I got it right, but obviously Ive missed something.

I thought that "a" would have been equal to 1/4. My reasoning is as follows:

First I simplified the equation, so instead of having:

5th root of 3^2,

it was wrote as

5th root of 9 (because 3*3 is 9)

I reasoned that since 3 is one exponent down from 9, I could instead massage the equation to help me solve for a.

The 5th root of 9, in my head, should be equal to the 4th root of 3.

And if the 4th root of 3 is equal to 3^a, then in my head we should arrive at "a" being equal to 1/4.

Can someone tell me what I'm doing wrong :D(5 votes)- It is when you assume that you can take the exponent down one. the fifth root of 3^2 would be 3^5/2. So your thought is that 5/2 = 4/1 which hopefully is obviously incorrect.(4 votes)

- but when the bases are same we add the exponents ? how 3a 5 =3 2

a = 2/5 ?(6 votes) - Does my way work:

3^a = 5root(3^2)

3^a = (3^2)^1/5

3^a = 3^(2/5)

a = 2/5(4 votes)- Yes, that works as well.(3 votes)

- what does 3^ mean if some one were to put this sympol " ^ " and a number in front of it.. what does that indicate?(2 votes)
- if someone puts the symbol "^" followed by a number that means to the power of...(3 votes)

## 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.