Cube root of a non-perfect cube

Let's see if we can find the cube root of 3,430. And if you're like me, it doesn't jump out of your mind what number times that same number times that same number-- if you have three of those numbers and you were to multiply them together-- would be equal to 3,430. So what I'm going to do is to try to prime factorize this to find all the prime factors of 3,430 and see if any of those prime factors show up at least three times. And that'll help us with this. So 3,430-- it's clearly divisible by 5 and 2, or it's divisible by 10. So let's do that. So first we can divide it by 2. It's 2 times-- let's see. 3,430 divided by 2 is 1,715. Then we can divide it by 5, as well. We can factor 1,715 into 5 and-- let me do a little bit of long division on the side here. So if I have 1,715, and I'm going to divide it by 5. 5 doesn't go into 1. It goes into 17 three times. 3 times 5 is 15. Subtract, you get 2, and then you bring down a 1. 5 goes into 21 four times. 4 times 5 is 20. Subtract. Bring down the 5. 5 goes into 15 three times, so it goes exactly 343 times. So 1,715 can be factored into 5 times 343. Now, 343 might not jump out at you as a number that is easy to factor. It's clearly an odd number, so it won't be divisible by 2. Its digits add up to 10, which is not divisible by 3. So this isn't going to be divisible by 3. It's not going to be divisible by 4, because it's not divisible by 2. It's not going to be divisible by 5. If it wasn't divisible by 3 or 2, it's not going to be divisible by 6. And now we get to 7. Usually when you see a nutty number like this that doesn't seem to be divisible by a lot of things, it's always a good idea to try things like 7, 11, 13. Because those tend to construct very interesting numbers. So let's see if this is divisible by 7. So if I take 343 and if I want to divide it by 7, 7 goes into 30-- it doesn't go into 3-- 7 goes into 34 four times. 4 times 7 is 28. Subtract, 34 minus 28 is 6. Bring down a 3. 7 goes into 63 nine times. 9 times 7 is 63. Subtract. We don't have any remainder. And I forgot to do that last step up here. 3 times 15 is 15. Subtract, no remainder. It went in exactly. So here, 343 can be factored into 7 and 49. And 49 might jump out at you. It can be factored into 7 times 7. So this is interesting. I can rewrite all of this here-- the cube root of 3,430-- now as the cube root of-- I'm just going to write it in its factored form-- 2 times 5 times-- I could write 7 times 7 times 7, or I could write times 7 to the third power. That captures these three 7's right over here. I have three 7's, and then I'm multiplying them together. So that's 7 to the third power. And from our exponent properties, we know that this is the exact same thing as the cube root of 2 times 5 times the cube root-- so let me do that in that same, just so we see what colors we're dealing with. So the cube root of 2 times 5, which is the cube root of 10, times the cube root-- and I think you see where this is going-- of 7 to the third power. Keeping track of the colors is the hard part. And the cube root of 10, we just leave it as 10. We know the prime factorization of 10 is 2 times 5, so you're not going to just get a very simple integer answer here. You would get some decimal answer here, but here you get a very clear integer answer. The cube root of 7 to the third, well, that's just going to be 7. So this is just going to be 7. So our entire thing simplifies. This is equal to 7 times the cube root of 10. And this is about as simplified as we can get just using hand arithmetic. If you want to get the exact number here, you're probably best off using a calculator.