Square Roots | Intro, Perfect Squares, and Simplifying | Math with Mr. J

In this video, I'm going to give a general overview of square roots. So we're going to start with, what are square roots? Then take a look at finding the square root of a perfect square. And then lastly, simplifying square roots. So let's start with, what are square roots? Now when we're looking for the square root of a given number, we need to think about what number multiplied by itself gives us that given number. That may not make any sense now, but after our examples, you'll see exactly what I mean. So let's jump into number 1, where we have a 3 and a 9. Before we talk about square roots, let's talk about squaring a number. Squaring a number means we have an exponent of 2. That means we multiply the number by itself. For example, 3 squared means 3 times 3. So let's do this. 3 squared means 3 times 3, which gives us 9, right? 3 squared equals 9. Now let's start with that 9 and do the opposite, or inverse, of squaring a number. And that's going to be taking the square root. So let's start with 9 and take the square root. So that's going to give us, well, the square root of 9. Let's think about what number multiplied by itself will equal the number under the square root symbol. This is the square root symbol there, also known as the root symbol or radical symbol. Well, we know 3 times 3 equals 9. So the square root of 9 is 3. Think about it. We know 3 times 3 equals 9. So a number times itself equals the number under the square root symbol. So the square root of 9 equals 3. Let's move on to number 2 and try another one. So 5 squared, 5 times 5, equals 25. Let's start with 25 and take the square root. So the square root of 25. What number times itself equals 25? Well, we know 5 times 5 equals 25. So the square root of 25 equals 5. 5 times 5 equals 25. A number times itself gives us the number under the square root symbol. So again, the square root of 25 equals 5. Let's move on to numbers 3 and 4. And we're just going to take the square root of these numbers. So for number 3, we have the square root of 4. Well, we know 2 times 2 equals 4. 2 times 2 equals 4. A number times itself equals the number under the square root symbol. So the square root of 4 equals 2. That's our final answer. And lastly, number 4, we have the square root of 36. Well, we know 6 times 6 equals 36. A number times itself equals the number under the square root symbol. So the square root of 36 equals 6. Now that we know a little bit more about square roots, we're going to take a look at more examples of finding square roots of perfect squares. So let's jump into it. Let's start with number 1, where we have the square root of 9. So we need to think about what number multiplied by itself will equal 9. So we know that 3 times 3 equals 9, right? 3 times 3 equals 9. So that's what I mean by a number 3 multiplied by itself gives us the number under the square root symbol. So the square root of 9 equals 3. And that's our answer. Let's move on to number 2, where we have the square root of 16. So we need to think again what number multiplied by itself equals 16. We know 4 times 4 equals 16. 4 times 4 equals 16. So a number multiplied by itself gives us the number under the square root symbol. So the square root of 16 equals 4. And that's our final answer. Let's move on to numbers 3 and 4, where we have 4 square roots for each. 3 of the 4 will have square roots of perfect squares, meaning they have a whole number answer. And 1 will not work out so nicely. Let's solve these and find which ones are the perfect squares and which one is not. So we'll start with number 3. And first, we have the square root of 81. So we need to think any numbers that multiply by themselves to get 81. Well, 9 times 9. So 81 is a perfect square. So the square root of 81 equals 9. So we get a whole number answer there. And that is a perfect square. So I will put a check there. Moving on to the square root of 26. So we need to think any numbers that multiply by itself to equal 26. Well, we know 5 times 5 equals 25. That's close, but not quite. And then 6 times 6 equals 36. So the square root of 26 is going to be somewhere between 5 and 6. It's not a perfect square, and it's not going to give us a whole number answer. So that's the square root that is not a perfect square within these four. Let's do the other two, which are perfect squares. So the square root of 49. 7 times 7 equals 49. So the square root of 49 equals 7. And this is a perfect square. Lastly, we have the square root of 100. And we know 10 times 10 equals 100. So the square root of 100 equals 10. And this is a perfect square. Lastly, number 4. We will start with the square root of 4. So is there anything that we multiply by itself to equal 4? Yes, 2 times 2 equals 4. So the square root of 4 equals 2. And this is a perfect square. Now we have the square root of 144. Well, 12 times 12 equals 144. So this is a perfect square. The square root of 144 equals 12. And it is a perfect square. Now we have the square root of 64. 8 times 8 equals 64. So this is a perfect square. The square root of 64 equals 8. And then lastly, we have the square root of 74. This is not a perfect square. So we're not going to get a whole number answer here. The square root of 74 is going to be somewhere between 8 and 9. Because 8 squared equals 64. And then 9 squared equals 81. So again, it's going to be somewhere between 8 and 9. This is not a perfect square. So we're not going to get a whole number answer. So there you have it. There are some examples of how to find the square root of a perfect square. Now that we have a better understanding of square roots and working with perfect squares, we're going to move on to simplifying square roots. So let's jump into that. In this section, we're going to take a look at simplifying square roots. And we have four examples that we're going to go through together in order to get this down. So let's jump into number one, where we have the square root of 20. Now 20 is not a perfect square. So we're not going to get a nice and clean cut whole number answer. So we need to simplify. We can do this by looking for factors of 20 that are perfect squares and then find their square roots. For example, we know that 4 times 5 equals 20. 4 and 5 are factors of 20. They go into 20, so to speak. 4 is a perfect square. So let's do this in order to simplify. The square root of 20 equals the square root of 4 times 5. Now the multiplication or product property of square roots lets us split this, meaning we can do this. The square root of 4 times the square root of 5. That's equivalent to the square root of 20. It's equal. We're not changing the value of the problem at all. Now we can take the square root of our perfect square, 4, and we end up with, well, the square root of 4 is 2. Bring down our square root of 5 because we cannot simplify that any further. So our simplified answer is 2 times the square root of 5 or 2 square root 5. Let's try another one and move on to number 2 where we have the square root of 32. So we need to think, are there any factors of 32 that are perfect squares? So any numbers that can go into 32, so to speak, that are perfect squares. If so, we can simplify. If not, it's already in simplest form. Well, we know that 16 times 2 equals 32 and 16 is a perfect square. So we can simplify. The square root of 32 equals the square root of 16 times 2. Let's split this. So we have the square root of 16 times the square root of 2. 16 is a perfect square. So the square root of 16 is 4. So we end up with 4 times the square root of 2 or 4 square root 2 or you can even say 4 root 2. Now 2 cannot be simplified any further. So this is our final answer as far as simplifying the square root of 32. Now I do want to mention we can take another path to get to that same answer for number 2. I'm going to try to squeeze it in to the left here. We know that 4 times 8 also equals 32 and 4 is a perfect square. So this is the other path we can take. And I'm going to start by splitting these just to make sure I have enough room. So we have the square root of 4 times the square root of 8. The square root of 4 is 2. Let's bring down our square root of 8. Now we can continue to simplify. We're not done yet because we have a perfect square within 8 as far as factors go. Because 4 times 2 equals 8 and 4 is a perfect square. So again we can continue to simplify. So we have 2 times the square root of 4 times the square root of 2. So the square root of 4 again is 2. So we end up with 2 times 2 times the square root of 2. Now the square root of 2 is simplified so we can't break that down any further. But we can multiply our 2 times 2. So 2 times 2 gives us 4. And then we have the square root of 2. So same exact answer but a different path. And that's okay. Both of those paths are correct. So keep that in mind. There may be different paths as far as simplifying a square root. Just always remember to check if you can simplify further. Let's move on to number 3 and try another one where we have the square root of 45. So think. Any factors that are perfect squares. Yes. 9 and 5 are factors of 45. 9 times 5 equals 45. And 9 is a perfect square. So the square root of 45 equals the square root of 9 times 5. Let's split this. So square root of 9 times the square root of 5. So the square root of 9 is 3. Bring down our square root of 5. That cannot be simplified any further. So we have 3 times the square root of 5 or 3 square root 5 or 3 root 5. Lastly, number 4, we have the square root of 75. So any factors that are perfect squares. Well, 25 and 3 are factors of 75. 25 times 3 equals 75. And 25 is a perfect square. So the square root of 75 equals the square root of 25 times 3. Let's split. So we get the square root of 25 times the square root of 3. Now the square root of 25 is 5. So we have 5. And then the square root of 3 is in simplest form. So we can't break that down any further. And this is our final simplified answer. Now I do want to mention that we always put the number before the square root symbol. As you can see in all four of our final simplified answers. So for example, if we were to put the 3 first. So the square root of 3 times the square root of 25. And get the square root of 3 times 5. You would want to rearrange this. So you have the number first and put the square root second. So you can see that we have the number then the square root. So that's common practice. And that's how you would want to leave your answers. So there you have it. There's a general overview of square roots. So what are square roots? Finding the square roots of perfect squares. And then simplifying square roots. Now I would suggest really knowing the first 12 perfect squares. This is going to help a lot as far as square roots go. And simplifying them. I have a pinned comment below with the first 12. So if you need some help with those, check that out. I hope that helped. Thanks so much for watching. Until next time. Peace.