If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Radicals and rational exponents — Harder example

Watch Sal work through a harder Radicals and rational exponents problem.

## Want to join the conversation?

• are there any other videos like this on Khan Academy. because I did not understand how to do this
• There are videos that can teach you exponent properties, which will make this easier for you.
• That was nice and all, but when I actually take the SAT I wouldn't know to do all that.
• when sal does it its so much easier but when i do it its a disaster
• lol me as well
(1 vote)
• I'll be honest I didn't understand this.
• At , he explains that 1/3 is the same as 3^-1. Is this the only way to figure this problem out, or is there another way?
• Yes, you could try approaching the 3^-1/5 differently by changing it to the fraction 1/[3^(1/5)]. Working with fractions is much more difficult than just working with the plain numbers though, so I would just use Sal's method because it saves you time.
• i solved for the first expression first, asking what multiples of 2 (2x) and 3 (3y) could be subtracted to equal 4 and came up with x = 5 and y = 2. (I'm guessing other combos might work, too, but i chose the lowest figures since we're going to use them as exponents in the second expression. So best to keep them small.)

so we have:
2x - 3y = 4
2(5) - 3(2) = 4
10 - 6 = 4

therefore x = 5 and y = 2 satisfy the first expression.

plugging into the second expression we have:

4^5 over 8^2
1,024 / 64 = 16

hopefully, the concept is sound and helps someone think this through in a diff manner.
• that's very helpful! could've used x=2 y=0 though.
works faster
• At , why don't you get 9 raised to the 1/5 since you are multiplying 3x3?
(1 vote)
• It sort of looks like that, if you squint and ignore the exponents. But 3^ (-1/5) ∙ 3^ (2/5) does not have any 3's in it really. It is `1 divided by the 5th root of 3` times the `5th root of three squared`.

What we have to do instead is use the exponent rules that say that
` xᵃ ∙ xⁿ = xᵃ⁺ⁿ `
and this applies even when a and n are fractions ("rational exponents")

Let's say that we have `9² ∙ 9^½` (just an example that is easier to deal with than the inverse of the 5th root of 3)
We cannot multiply 9 times 9 to get 81 and then do the exponent addition, which would be 2 + 1/2 = 4/2 + 1/2 = 5/2 as the exponent. Then we would have 81^5/2
That number would be 59049

Instead, 9² = 81 and 9^½ = 3, so `9² ∙ 9^½` = 81 ∙ 3 = only 243 and that is a lot less than 59049

When we apply the exponent rules for products of powers of the same number, we don't multiply the base, we just add the exponents

In this math problem, we have exponents of -1/5 + 2/5 which result in 1/5 as the simplified exponent
So the answer is 3^1/5

By the way, that is the fifth root of 3 which is ⁵√3
• I did not understand this and to worry more this will be in my sat tommorow
• You can look for other videos or tell us the part where you didn't get it.
Maybe in your next test try not to leave topics for the very end.
• oh god
• Damn i lost my last brain cell because of this question 🥲

## Video transcript

- [Tutor] We're told if 2x - 3y is equal to four, what is the value of four to the X power divided by eight to the Y power? Pause this video and see if you can figure this out. All right, so at first this looks a little bit tricky. You're like, how do I manipulate what I have here on the left to get what I have here on the right? But another way to approach that is to say, look, this thing on the right looks a little bit suspicious, four and eight they aren't... Eight isn't a power of four, but we know that they are both powers of two. And so, maybe we can re-express four as a power of two, and we can re-express eight as a power of two. And maybe if we algebraically manipulate that this might show up, so let's see what happens. So, I'm just going to rewrite everything. So, we have four to the X power over eight to the Y power. Now, as I just mentioned, four is the same thing as two squared? So, we can rewrite this as two squared and then that's to the X power over instead of eight we know that eight is the same thing as two to the third power, and all of that to the Y power. Now, if we know, we know already from our exponent properties, and if this is unfamiliar to you, you can review it on Khan Academy. If you raise something to an exponent and then raise that to another exponent, that's equivalent to multiplying the exponents. So, this is going to be equal to, and I'm gonna get a new color here. This whole numerator is going to be equal to two to the two times X power or two to the two X power and that's going to be divided by, and then this entire denominator right over here, it's going to be two to the third to the Y. So, it's going to be two to the three times y power. Two to the three Y power. Now, we have the same base and we can use other exponent properties. You might recognize that if I have A to the X over A to the Y, this is the same thing as A to the X minus Y. And we explain the intuition of that in other videos on Khan Academy, but we can use that property right over here. We have the same base, and so, this is going to be equal to two that same base to the 2x - 3y power minus we have our 3y over here minus 3y power. And so, this whole thing has been remanipulated or manipulated to be two to the 2x - 3y power and say, where do I go from here? Well, we just have to remember they told us that 2x - 3y is equal to four. So, all of this business is equal to four. So, it's two to the fourth power. Well, we're in the homestretch now deserve a little bit of a drum roll. This is equal to 16 and we are done.