ALEKS Math Placement Test Practice Problem | Cube Roots Explained Step-by-Step

Let's talk about Alex Math. Now, if you want to do really well on the Alex Math placement exam, this should be a pretty easy question for you to answer without the aid of a calculator. Alright, so here is the problem. We want to figure out what the cube root of 80 is equal to. Now, I'm going to solve this in just one second, but if you think you know the answer, go ahead and put that into the comments section. I'm going to walk through the step-by-step solution right now. Now, you may not necessarily need to understand something called rational exponents, but it's very useful when we're talking about how to solve this problem. Of course, I'm going to go through all of this right now. Before we get into this problem, let's just do a quick review between the difference of a square root and a cube root. So, let's take a look at this problem. Here we have the square root of 16. So, what are we looking for? Well, we're looking for a number such that when we multiply it by itself, we get back to 16. Now, of course, we know the answer is 4, so 4 is the square root of 16. Now, in this symbol in mathematics is the square root symbol, but it's also called a radical. Alright, so we can have things other than square roots, of course, like a cube root. Now, when we write a square root, technically there's a little 2 right here, but we don't really write that. Okay, but you can see that in contrast to this cube root problem right here. Now, just a quick comment. Here we have the square root of 16 is 4. Now, if you're thinking about positive and negative 4, that would be in this situation where you have like a quadratic equation. So, these would be the roots or solutions to this quadratic equation, but typically when you just have a number like the square root of 16, you're going to write the positive value only. That's called the principal square root. Okay, so this is a square root. Now, let's take a look at this cube root problem. So, here we have the cube root of 27. So, what are we looking for here? Well, remember when we had a square root, it was a number times itself. So, we're talking about two values, right? The same number times itself that gets back to this number. So, the same thing holds true here, right? So, we're trying to find the cube root of 27. So, we're looking for a number times itself three times the same number that gets us back to 27. Now, of course, the answer is 3 because 3 times 3 times 3. 3 times 3 is what? That's 9. 9 times 3 is 27. So, the cube root of 27 is 3. Okay, so that's just a quick review of square roots and cube roots, and now we're going to talk about rational exponents. Okay, so I can think of these problems, these same problems this way. Alright, so the square root of 16 is the same thing as 16 to the 1 half power. Now, this is what we call a rational exponent in mathematics. Now, that word, this word rational means what? Well, effectively, it means a fraction that is made up of integers, right? So, a fraction with kind of whole number values. Now, of course, I can kind of get into more about rational numbers, but hopefully you understand what that means. Now, when we're talking about rational exponents, what we need to do is the following. Okay, so you remember there's a little 2 here when we have our radical symbol or square root symbol. This is going to become the denominator, and 1 is always going to be the numerator. So, 16 to the 1 half power is the same thing as the square root of 16. Now, if you go into your calculator and just test that, you can just take your 16 and use your care key or your Y to the X key, put parentheses, 1 half, and you'll see the answer is 4. Now, likewise, the cube root of 27 is what? Well, I can express this using rational exponents as well. So, this is going to be 27 to the 1. Remember, 1 is always the numerator, and the denominator is the root. Okay, so 27 to the 1 third power is the same thing as the cube root of 27. Okay, so this is going to help us out because oftentimes when you're dealing with radicals, square roots, cube roots, etc., working with rational exponents makes our life a lot easier. So, let's go ahead and take the next step to solve this problem. If you're going to be taking the Alex Math placement exam, you don't want to take a chance of getting into the wrong class in college because of weak math skills. What you want to do is strengthen those math skills. So, instead of guessing on math questions, you are answering them confidently. So, make sure to check out my full Alex Math test prep course. It is extremely comprehensive with full step-by-step lessons and thousands of practice problems. So, if you want to check it out, just follow the link in the description. So, now that we know something about square roots and cube roots and rational exponents, what we need to understand to solve this problem is how to simplify a radical. And let's take a look at this simple problem right here. Remember, in algebra, this symbol right here is called a radical. So, if you're like in an algebra course and you're studying square roots and cube roots, well, you're probably in a chapter or unit called radical equations, radical expressions, etc. Okay, so let's talk about how to simplify the square root of 20 without using a calculator. Alright, so how do we do this? Well, the main idea is the following. We want to factor the number underneath the radical. In this case, it's a square root. And we want to factor this into two or more numbers. Now, what type of numbers could we use as factors of 20? Well, of course, 2 times 10. These numbers here are factors of 20, but so is 4 times 5. Now, the main idea here is that we want to look for something called perfect square factors, right? Perfect square factors. Now, a perfect square are numbers like this, 4, 9, 16, 25, etc. Now, why are these perfect squares? Well, because when we take the square root of these values, we get nice, lovely, perfect numbers like 2. So, the square root of 9 is 3. The square root of 16 is 4. The square root of 25 is 5, etc. Okay, so these are perfect squares, and we want to try to identify perfect squares as factors of this number. All right, so here we have the square root of 20. Now, of course, we have different factors of 20, 2 and 10, and 4 and 5. But are any of these here perfect squares? Well, of course, 4 is a perfect square. Okay, so let's go ahead and talk about why this is going to be critical. So, we can think of the square root of 20 in terms of its factors, right? So, I can think of the square root of 20 as equal to the square root of 2 times 10. Now, these are the factors of 20, but this is not going to help me out so much, okay? But if we think of the square root of 20 as the square root of 4 times 5, and you can see here I have this perfect square. Well, this is going to be the key to simplifying this square root. Okay, now why is that? Well, we have a property of square roots and radicals, and it goes like this. So, when you have a square root or a radical, you can look at that value in terms of its factors. So, the square root of 20 I can write as the square root of 2 times 10, but here is the key. We have a property, a rule, that allows us to break up this one big square root into separate square roots of the factors. So, instead of the square root of 2 times 10, we can write this as the square root of 2 times the square root of 10. Okay, now why is this important? Well, if we think of the square root of 20 not as the square root of 2 times 10, but as the square root of 4 times 5, I can break up this big square root into two individual square roots. So, now this is the square root of 4 times the square root of 5, and now we can actually find the square root of 4, which of course is 2. So, our final answer here becomes 2 times the square root of 5, and this is the simplified version of the square root of 20. Okay, so we're basically going to do the same thing with this problem, the cube root of 80, as we did with the square root of 20, but instead of thinking about perfect squares, we need to think about perfect cubes. So, here is 80. Now, what is a perfect cube? Well, let's take a look at the one example that we already saw, and that is 27. When we take the cube root of 27, the answer is what? Well, that is 3. Matter of fact, let me write that right here. So, 27 is a perfect cube because the cube root of 27 is 3. But what is the cube, or what is a perfect cube before 27? Well, how about 8? Because when we take the cube root of 8, we get 2. Okay, so hopefully you can see here that 8 is going to be a great factor to use with this number here, right? We have the cube root of 80, and 8 is, of course, a factor of 80. So, what we want to do is break up 80 in terms of a perfect cube, and we can easily use 8. Okay, so let's go ahead and take a look at that work right now. So, instead of 80, we're going to think of 80 as 8 times 10. Okay, so let's go ahead and use the same properties here. Again, when we're dealing with radicals, we're talking about this symbol here. So, it's not just square roots. It could be a cube root, a fifth root, etc. Okay, so the cube root of 80 we want to think of as 8 times 10. Now, of course, 8 we can express as the power 2 cubed, right? Because 8 is 2 times 2 times 2. Now, that's not really super critically important as long as you realize that 8 is a perfect cube. But here is the key part of this problem. We can express the cube root of 80 as the cube root of what? 8 times 10. Now that we understand this property that we can break up these radicals in terms of its factors, the cube root of 8 times 10 is the same thing as the cube root of 8 times the cube root of 10. And, of course, the cube root of 8 is what? Well, that is 2. So, our final answer here is going to be 2 times the cube root of 10. All right, so I hope this video helped you out if you are preparing for the Alex Math placement exam. And, again, if you want more step-by-step practice like this, make sure to check out my full Alex Math test prep course. You can get started right away by following the link in the description.