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Sal simplifies 3√(500x³) as 30x√(5x). Created by Sal Khan and Monterey Institute for Technology and Education.
Video transcript
What I want to do in this video is simplify this expression 3 times the principal root of 500 times "X to the third" and take it into consideration some of the comments that we got out on YouTube that actually uh, gave some interesting perspective on how you can simplify this. So this is a quick review of what we did in the last video, we said that this is the same thing as 3 times the principal root of 500, and I'm gonna do it a little bit different than I did in the last video, just, just so to make it interesting. This is 3 times the principal root of 500 times the principal root of "X to the third." And, 500, we can rewrite, because 500 is not a perfect square, we can rewrite 500 as 100 times 5. Or even better, we can rewrite that as 10 squared times 5... 10 squared is the same thing as 100. So we can rewrite this first part over here 3 times the principal root of 10 squared times 5, times, times the principal root of x squared times x. That's the same thing as x to the third. Now the one thing I'm gonna do here... actually I won't talk about it just yet, of how we are gonna do it differently in the last video. This radical, right here, can be rewritten as... So this is gonna be three times the square root, or the principal root, I should say, of 10 squared times... the square root of 5. If we take the square root of the product of two things, it's the same thing as taking the square root of each of... them, and then taking the product. And, so in this over here is going to be times the square root of where the principal root of x squared times... the principal root of "X." And, the principal root of 10 squared is 10, and then what I said in the last video was the principal root... of "X" squared is going to be the absolute value of "X." Just in case, just in case, if "X" itself is a negative number. So, then if you simplify all this, you get 3 times 10, which is 30 times, and I'm just gonna switch... the order here, times the absolute value of X, and then you have this square root of 5, or the principal root of 5, times the principal root of X. And this is just going to be equal to the principal root of 5x. Taking the square root of something, and multiplying that times the square root of something else... is the same thing as just to taking the square root of 5x. So, all of this simplified down to 30 times the absolute value of X, times the principal root of... 5x. And this is what we got in the last video. And the interesting thing here is, if we assume we're only dealing with real numbers, the domain... of X, right over here, the X's that will make this expression define in the real numbers... then, X has to be, greater than or equal to 0. So, let me, so maybe I could write it this way. The domain, here, is that X is any real number, greater than or equal to 0. The reason why I said that is that you put a negative number in here, and you cube it, you're... gonna get another negative number. And, then it doesn't make it, at least in the real numbers, you won't get an actual value. You'll get a square root of a negative number. So, if you make this, if you assume this right here, we're dealing with the real numbers... we're not dealing with any complex numbers, when you hope their not complex numbers, then... you can have a, you can expand the domain more broadly. If you're dealing with real numbers, you can say that X is going to be greater than or equal to... 0. And, then, the absolute value of X is just going to be X, 'cause it's not going to be... a negative number. And, if, so assuming that the domain of X is, er, if this expression is going to be invaluable, or it's... it's going to have a positive number, that this can be written as 30X times the square root of... ... of 5X. If you had the situation, where we're dealing with complex numbers, then you would... So numbers that were, and if you don't know what a complex number is, or an imaginary number, don't... worry too much about it. But, if you were dealing with those, then you would have to keep the absolute value of X... there, because then this would be defined for numbers that are less than 0.