The Best of Simplifying the Cube Root of a Number

what's up guys in this video I'm going to show you how to simplify the square root of a number and I'm going to do that by working through five different examples I have hand selected to help you understand the process I hope you enjoy so when taking the cube root of 54 um as as you guys remember from when I was talking about square roots I really don't like using prime factorization all right to me it's very time um consuming however it's a process that always always works always always works so it's very helpful in that regard all right however I always prefer just like with square roots is to find square numbers that you can divide to so when we're talking about cube roots you want to find Cube numbers and what's nice about simplifying Cube numbers is there's not that many cubed numbers that you're going to be working with 2 cubed is eight three cubed is 27 4 cubed is 64. 5 cubed is 125. 6 cubed is 216 but unless you're doing some really really high numbers these are pretty much basically going to be your major numbers that you guys are going to be dealing with all right hence why I already have these memorized because we don't really get up to 7 cubed 8 cubed 9 cubed we do for square roots but not really for the cube numbers so when I'm looking for this if I want to simplify cube root yes we could prime factor 54. there's not a problem with doing that however I always think it's simpler or easier to be able to determine which of these numbers is any of these numbers divis um divide into 54. yeah 27 and it goes in there twice so what I can do is rewrite this as a cube root of 27 times 2. now I know the cube root of 27 that's 3. times the cube root of two and I can't simplify the cube root of 2 so that's just going to remain as 2. okay so in this example ladies and gentlemen um when we're dealing with square roots you know you guys are pretty familiar with square notes um uh you know what number multiplied by itself we know that the square root of 16 is 4. but when we get into cube roots now we're thinking of what numbers multiplied by themselves well we know that three two times two times two is eight three times three times three is going to be 27. so we can't take this the square the cube root of 16. however we can simplify so my recommendation to you is when you're doing this is again you can use the factor tree because guys it doesn't matter if you have your radicand use a factor tree to write it as prime factors because it doesn't matter if it's the cube root the fifth root the sixth root you're always going to want to come from the same factor tree so I break this down into eight times two I break that down into four times two and that broken down to 2 times 2. so what I want you guys to understand is that the cube root of 16 is the same thing as 2 times 2 times 2 times 2. do you guys agree with me so if you take the factor tree and then write your number as a prime factorization now to take the cube root you need to take pairs or you need to group how many of the twos three so the cube root of 2 times 2 times 2 is just two and then you still have a 2 left under your radical so in this case what I'm trying to do is identify what number multiplied by itself three times gives us negative 27. well ladies and gentlemen I kept these numbers up here because I want you guys to write these down here is a list of square numbers these are all numbers we can take the square root of 2 times 2 is 4. 3 times 3 is 9. 4 times 4 is 16. 5 times 5 is 25. do you guys see that these are what we call Cube numbers these are numbers you can take the cube root of because 2 times 2 times 2 is 8 3 times 3 times 3 is 27. so what you notice is what number multiplied by itself three times gives us negative 27. well we only have two options it's either negative 3 times negative 3 times negative 3 or 3 times 3 times 3 right and it has to be negative 3 because that gives you 3 and this is going to give you negative or negative 27. so therefore ladies and gentlemen a couple things I want to first talk about is talking about square roots versus cube roots all right now when we're go ahead and looking at determining how to take the cube root because this is one thing we have not covered yet in this class cube root is going to be something very similar to the square root all right and there's two different ways that we learned how to do the roots and the first way to learn how to figure out the roots was always to apply the prime factorization of a root right yes it is it's just a hard pull for a lot of people so when we're looking into completing the prime factorization it doesn't matter if you're doing square root it doesn't matter if you're doing cube root you can always start by applying the prime factorization of that number so if I was going to do that I would just take 48 and ladies and gentlemen again the prime factorization we can break that down you know we know it's 24 times 2 then you can do 6 times 4. three times two two times two so therefore ladies and gentlemen if I was going to do this for the square root I would do three times two times two times two for the cube root I would do three times two times two oops I forgot one right now the difference between the square root and the cube root for the square root we always want to pull out pairs right because it's Square so there's two of them so I could pull out one pair and two pairs so I'd have two times two times square root of 3 which is 4 square root of 3. however for the cube root Harvey let's take a random guess if for the square root I'd take out pairs of Twos for the cube root I would want to take out pairs or groups of three not three so therefore that's going to be 2 times the cube root of 3 times 2 which is left over which would be six does that make sense okay so ladies and gentlemen if I throw one with is the fourth root how many will you want to take out for the fourth root four so the exact same process of prime factorization is going to be exactly the same in this example ladies and gentlemen what I have is I have the cube root of negative 64. Jake not right now so if I'm going to look into taking the cube root of negative 64. again guys what we're looking into is what number multiplied by itself what exact same number multiplied by itself is going to give us negative 64. but remember that number has to be multiplied by itself three times now there's a couple different ways to do this and we're going to get into some more com um some more complicated numbers um and assume but the base thing I always like to do is when I'm looking into determining cube roots or square roots is always small start with the smaller numbers 2 times 2 times 2 is 8. 3 times 3 times 3 is 27. 4 times 4 times 4 is actually yes negative 64. so I could do the cube root of four times four times four or since it has to be negative 64. I see that 4 times 4 times 4 gives me positive 64. but remember the cube root is saying what number multiplied by itself three times gives me a negative 64. and that ends up being negative four do you guys see how negative 4 multiplied by itself three times is going to give you negative 64. so the answer is just negative 4.