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.

Main content

Radicals and rational exponents — Harder example

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

Want to join the conversation?

  • aqualine seed style avatar for user Gabriel Babuch
    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?
    (10 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Sachin Raghavan
    I'll be honest I didn't understand this.
    (8 votes)
    Default Khan Academy avatar avatar for user
  • starky seedling style avatar for user Kameron Wilson
    are there any other videos like this on Khan Academy. because I did not understand how to do this
    (1 vote)
    Default Khan Academy avatar avatar for user
  • starky tree style avatar for user mel.bookwriter
    At , why don't you get 9 raised to the 1/5 since you are multiplying 3x3?
    (1 vote)
    Default Khan Academy avatar avatar for user
    • purple pi purple style avatar for user doctorfoxphd
      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
      (15 votes)
  • piceratops sapling style avatar for user Sam Saleh
    I did not understand this and to worry more this will be in my sat tommorow
    (6 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user shakibakalemzai
    why at the end you got 16 how ?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • female robot ada style avatar for user bear.gummie
    Fractions are my weakness, I usually end up with the wrong sign.
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Paul Armstrong
    I did it differently. 3^-1/5 = (1/3)^1/5 and (1/3)^-2/5 = (1/(1/3)^2/5), which in turn is equal to 3^2/5. So, (1/3)^1/5 X 3^2/5 = ((3)^2/5)/((3)^1/5). When you divide 2 powers with same root subtract exponents, so it equals 3^1/5
    (3 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Portillo, Sophia
    At , If the base, 2, was not the same for the numerator and denominator (eg. 2^2x/ 3^3y) how would you solve it since you can't use the property of a^x/a^y?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • mr pants purple style avatar for user Aulani Henry
    So, when you come to your final equation of 3(with the exponent of)-1/5 and 3(with the exponent of)2/5...You subtracted the two exponents, or was one of them still considered a negative?
    (2 votes)
    Default Khan Academy avatar avatar for user

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.