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Simplifying higher-index root expressions

How to rewrite a radical with variables in it as an exponent and then simplify it using exponent properties. In this example, we simplify ∜(5a⁴b¹²). Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user J.M._Behrens
    What's the difference between a radical and a surd?
    (34 votes)
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  • leaf blue style avatar for user Wang Ran
    based on previous video shouldnt the result be 4^√5|a||b|^3 ? since both a and b can either be positive or negative number...?
    (18 votes)
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    • hopper cool style avatar for user Ronald Perez
      no, ask yourself, if both a and b can be either negative or positive, why should we force it into an absolute and make it only a positive-holding variable? you have to remember variables can hold any number, it's only a representation of what it's supposed to be holding, if a = -1, then we change it by forcing the equation to b positive one, thus the statement is false.
      (0 votes)
  • mr pants teal style avatar for user Alex
    Just a small quick question, would raising a number to the fourth power be 'hypercubing' in shorthand, like how anything raised to the second power is squared, and raising anything to the third power is cubing. So you know, hypercube is a 4-d cube, so just as ^2 is squaring, and in 2 dimensions, and ^3 is cubing, and in 3 dimensions, maybe ^4 is hypercubing, and in 4 dimensions.
    (4 votes)
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    • aqualine ultimate style avatar for user Selena Liu
      I suppose, although I've never seen anyone use that. Following your logic, I guess something to the 1st power would be "lining" and something to the 0th power would be... "pointing"? I don't know, it's up to you to decide. Just make sure the person you're talking to knows about "hypercubing" before you use it.
      (8 votes)
  • aqualine seed style avatar for user jamiraljones4218
    ok so when i ask questions about what i dont get i know i dont get something i just don,t know what i don,t get do you get it - vampier119
    (6 votes)
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  • blobby green style avatar for user Dayna Rodriguez
    Can the answer also be (ab^3)^4
    ^4 square root of 5?
    (5 votes)
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  • blobby green style avatar for user Lizeth Nunez
    this is so hard i dont get it
    (2 votes)
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    • male robot donald style avatar for user David
      Hey Lizeth,
      What specifically are you having trouble with? By the way, don't worry if you're not getting it now. I used to hate math A TON and thanks to several great teachers (and Khan Academy), I now understand a lot of the stuff that I used to hate. Just make sure you keep asking for help whenever you need it. If you keep whacking at it (with the right help from other people), you'll understand it eventually.
      (4 votes)
  • leaf green style avatar for user lakyrahtaylor09
    Hi! I am really confused. can somone help me with the quetion that says the square root of 28 by simplyfying each expression because they are all confusing to me. thank you
    (2 votes)
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  • blobby green style avatar for user Michelle Page
    Why doesn't the fourth root of five come first? Or does the ordering not matter.
    (1 vote)
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    • piceratops ultimate style avatar for user abassan
      The rule is that you always put the radical at the end of an expression.
      I think the reason is that, this way it's clear what is inside the radical and what is outside the radical. People have a tendency to make the top of the radical a little too long or too short and then you can't tell which term are inside. By putting the radical last, we know that everything that comes after the radical is inside.
      (4 votes)
  • purple pi purple style avatar for user lindsell4
    Shouldn't the answer be written with a rational exponent (rather than leaving a radical in the answer)? That is, 5 to the 1/4 power x a x b to the 3rd power?
    (2 votes)
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  • piceratops ultimate style avatar for user Soldier of Christ
    What are rational exponents?
    (2 votes)
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Video transcript

Rewrite the radical expression using rational exponents and simplify. So here we have the fourth root of 5 a to the fourth b to the twelfth power. The key thing to realize here is the fourth root of something is same thing as something to the one fourth power. Or in particular, or in general, the nth root something is the same thing as that something to the 1 over n power. So we can just apply that over here: the fourth root of all of this is equal to 5 a to the 4th, b to the 12th power, all of that to the one fourth power. And then we also know if we take the product of things, and then raise them to some exponent, that's the same thing as raising each of the terms in the product to the exponent first, or each of the things that we're taking the product of to that exponent, and then multiplying. So let's do that. So 5, so this is the same thing as 5 to the 1/4 power times a to the fourth to the 1/4 power times b to the 12th to the 1/4 power. Now 5 to the 1/4, I don't know what that is, so I'll just keep that as the cube root, well, we could leave it as 5 to the 1/4, and that's not not simplified. or we can just rewrite it again as the fourth root of 5. a to the fourth to the 1/4 power: if your raise something to a power and then another power, and raise that to another power, that's equivalent of raising a to the four times 1/4 power. So let me just write that down. This is, so times a to four times 1/4 power. And then finally, this right over here, using the same exact exponent property, this is b to the 12th times 1/4 power. So all of this simplifies to, and I'll change the order here, so you have the fourth root of 5, and then you have a to the fourth times 1/4 power so that's just, this simplifies to a to the first power which is really just the same thing as a. So that's just a. And then we have b to the 12 times 1/4 power. Well, 12 times 1/4 is just three. So that's b to the third power. So it's a, b to the third power times the fourth root of 5.