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### Course: College Algebra>Unit 13

Lesson 1: Rational exponents

# 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?
• 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?
• If 3^a is raised to the 5th power, shouldn't it be 243^5a and not 3a^5a?
• 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.
• 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...
• Yes you are correct because you still would have common bases equal to each other.
• couldnt you have just gone 3^2/5 instead of all that other stuff?
• Yes, your approach works. Remember, this is an intro video and other students may need an alternate approach to understand the relationships.
• 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 ;)?
• Your solution is correct. In fact, it's the same solution as alexa.pomerantz's above.
(1 vote)
• 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?
• 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.
• 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
• 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.
• but when the bases are same we add the exponents ? how 3a 5 =3 2
a = 2/5 ?
• Does my way work:
3^a = 5root(3^2)
3^a = (3^2)^1/5
3^a = 3^(2/5)
a = 2/5