In this video, we're going to focus on simplifying square roots. We're going to start with some basic examples, and then gradually, they're going to get harder. So consider these four examples.
How would you simplify it? Let's start with the first one. What is the square root of 49? What two identical numbers, when multiplied, will give you 49? 49 is 7 times 7. So it turns out that the square root of 49 is 7. What about the square root of negative 25?
Well, this won't give you a real number, but this will give you an imaginary number. What you can do is break it up into 25 times negative 1. And it's important to understand that the square root of negative 1 is the imaginary number i. Now what is the square root of 25? What two numbers multiplied, what two identical numbers when you multiply them will give you 25?
We know that 5 times 5 is 25. So the square root of 25 is 5. And the square root of negative 1 is i. So this gives us the imaginary number 5i. Now what about negative square root 81? This time it's a little different than the previous example.
The negative is on the outside. So we're not going to get an imaginary number, but we're going to get a real number. That negative sign will remain on the outside.
So what is the square root of 81? What number times itself is equal to 81? Now we know that 9 times 9 is 81. So the square root of 81 is going to be 9. So the answer that we get in this case is negative 9. Now what about negative square root of negative 64? What's the answer there? If you have a negative sign inside a square root, it's best to remove it by writing the square root of negative 1 next to it.
So we can replace this with i. Now the square root of 64 is 8, because 8 times 8 is 64. And so the final answer is going to be negative 8i. Now sometimes you may have to simplify square roots that don't contain perfect squares. For example, how can we simplify the square root of 18 and the square root of 75? Now you need to understand what are perfect squares.
1 is a perfect square because 1 times 1 is 1. 4 is a perfect square. 2 times 2 is 4. 9 is a perfect square because 3 times 3 is 9. 4 times 4 is 16, 5 times 5 is 25, 6 times 6 is 36, 7 squared is 49, 8 squared is 64, 9 squared is 81, 10 squared is 100. So these are known as perfect squares. Because if you have the square root of any of these numbers, like let's say the square root of 36, it's going to simplify to an integer. But now 18 and 75 are not included in this list. So how do we simplify the square root of 18 and the square root of 75?
What would you do here? One thing that you could do is you can break up the number 18 into two smaller numbers, one of which contains a perfect square. So what perfect square goes into 18?
18 is divisible by 9. So what I would do is write the square root of 18 as being the square root of 9 times the square root of 2 because 9 times 2 is 18. And now at this point, we know what the square root of 9 is. The square root of 9 is 3. And so the final answer is 3 square root 2. And so that's a simple way in which you could simplify square roots. Let's do the same for the square root of 75. So what is the highest perfect square that goes into 75? 25 goes into 75. 75 divided by 25 is 3, so we can write 75 as being 25 times 3. and the square root of 25 is 5. So the square root of 75 simplifies to 5 square root 3. Now let's work on some more similar problems for the sake of practice. Try these two, the square root of 12 and the square root of 48. Feel free to pause the video as you work on those two examples.
So the highest perfect square that goes into 12 is 4. So we can write 12 as 4 times 3. And the square root of 4 is 2. So this is going to give us 2 square root 3. Now what perfect squares can go into 48? 48 is divisible by 4, and it's also divisible by 16. So what do you do in this scenario? If you have multiple perfect squares that can go into a number, pick the highest one, in this case 16. 48 divided by 16 is 3. So we can write 48 as being 16 times 3. And the square root of 16 is 4. So the answer is going to be 4 square root 3. Try these two problems. 4 square root 98 and also 7 square root 80. Now which perfect square goes into 98?
49 goes into 98. And you could write 98 as being 49 multiplied by 2. Now what is the square root of 49? We know the square root of 49 is 7 and now we need to multiply 4 by 7 which is 28. So the final answer for that problem is 28 square root 2. Now what about for the next one? What perfect square goes into 80?
80 is divisible by 16. If you take 80 and divide it by 16, you're going to get 5. So you can write 80 as being 16 times 5. And the square root of 16 is 4. So now we have 7 times 4, which is 28. And so this is going to give us 28 square root 5. And so that's it for that problem. Now what would you do if you have a problem that looks like this? 3 square root 18 plus 5 times the square root of 72 minus 4 times the square root of 32. How would you simplify this expression?
Now, it's important to understand that at this point, we cannot add the coefficients of the radicals, because right now, what's inside the square root are different. But if they were the same, we could. For instance, to illustrate that, we can't say 3x plus 5y is 8xy. That doesn't work. However, we can add like terms.
So we could say that 3x plus 5x is 8x. So the only way we can add the coefficients is if the radicals are the same. For example, if I had 4 square root 3 plus 5 square root 3, because the radicals are the same, I can add the coefficients. 4 plus 5 adds up to 9. So what I need to do in this problem is simplify the square roots.
In such a way that all of these numbers inside the square root that remains will be the same. So 18, we can write that as 9 times 2. The perfect square that goes into 72 is 36. 72 is 36 multiplied by 2. And 32, we can write that as 16 times 2. Now the square root of 9 is 3. And the square root of 36 is 6. And the square root of 16 is 4. So now I can multiply 3 times 3. That's going to give me 9. 5 times 6 is 30. and 4 times 4 is 16. So now at this point I can add the coefficients. 9 plus 30, that's 39, minus 16, that's going to be 23. So the final answer is 23 square root 2. And so that's how you could simplify problems like this.
You need to simplify each square root until... you get identical square roots, and then you can add the coefficients. So I'm going to stop it here today.
That's it for this video. Hopefully you found it useful. If you did, feel free to subscribe to this channel.
Thanks again for watching.