About this video
Sal solves 3^a = fifth root of 3^2 by raising both sides to the fifth power, using (3^a)^5 = 3^(5a), and matching exponents to get a = 2/5. He then uses the result to show why a rational exponent like 2/5 means taking the fifth root of the base squared.
You'll be able to
- solve an exponential equation by raising both sides to a power and matching exponents.
- use (x^a)^b = x^(ab).
- explain why 3^(2/5) is the fifth root of 3^2.
Key words
- fifth root
- The number that, multiplied by itself five times, gives the original number.
- power of a power
- (x^a)^b = x^(ab): raise a power to a power by multiplying the exponents.
- same base
- If 3^m = 3^n, the exponents must be equal, so m = n.
- rational exponent
- A fraction exponent; in 3^(2/5) the denominator 5 is the root and the numerator 2 is the power.
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.
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