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Welcome to Algebra I: Simplifying Square Roots of Whole Numbers

17 minutes

Hey, guys. Welcome to Algebra I. Today's lesson will focus on simplifying square roots of whole numbers. All that you've learned about perfect squares and square roots, it's all gonna come in handy in this one. Ready? Let's go.

(Describer) Standing at a screen, she holds a stylus.

Now, before we get into the meat of this lesson, I've got to jog your memory about perfect squares and square roots, okay? Let's look at these two.

(Describer) ...on the screen.

The first one's telling me to find the square root of 9. That little symbol over the 9, it's technically called a "radical." It's often called a "root," sometimes a "square root." That little three in there, it's a "cubed root." Get to those later. This one's asking me to find the square root of 9, which essentially means what number times itself equals 9? You think back. You run down numbers. It's 3. Because 3 times 3 is 9. So, I'd say, "All right. The square root of 9 is 3"...

(Describer) She writes.

I'm gonna write a little "because" here... "3 times 3 equals 9." You don't have to write the "because." I want you to know why the answer is 3. Okay?

(Describer) The next one.

So, 36? You think, what number-- This is asking you to find the square root of 36. What number times itself equals 36? You run down the list. You're thinking. Maybe you're checking in your calculator. And it's 6...

(Describer) She writes.

because... 6 times 6 is 36. Okay? That's essentially what it means when you take a square root. You figure out what number times itself is gonna equal whatever that number is under your radical. Look at this next one with me. The square root of 50. Hmm.

(Describer) She changes the screen.

Okay. When you're thinking about that--you're thinking, you're thinking, you're thinking. You're like, "Wait. There's got to be something." There really isn't. There's no whole number that you can multiply times itself and it'll equal 50. There's not. Sometimes people say, "Yeah, there is. It's 25, right?" What'd they do if they told me it's 25? They just divided 50 by 2. There's no number you can multiply by itself and get 50. When you have to take the square root of a number that's not a perfect square, you have to simplify the radical. You represent it in another way that's simpler, but you don't get a whole number for your answer. To know exactly how to do that, you have to be familiar with perfect squares. We know numbers go on for infinity. They have technically no beginning and no end, but there are some that we commonly use. Let's generate that list. I'm gonna ask you to give an answer. 1 times 1. That's 1. That's a perfect square. 2 times 2 is 4. 3 times 3? That's 9. Okay, I'm squaring these. Four squared? What's 4 times 4? Sixteen. Five squared? That's 25. Six squared, 6 times 6? Thirty-six. Seven squared? Forty-nine. Eight squared? Sixty-four. Nine squared? Eighty-one. Ten squared? A hundred. You're probably like, "When is she gonna stop?" In a second. Eleven squared? 121. Just for kicks, it's not really common, but 13 squared? It's actually 169. If you can be familiar with these, at least up to 11 squared, to 121, you'll be good working your way through these. It may seem overwhelming, but you're gonna work with these numbers so much that it's gonna become second nature to you. But while we're learning it, you might want to write a list, so while we're working with these problems, you have a list of the most commonly used perfect squares. Now that we have this list, let's go back to that square root of 50. Now, how we simplify this, let's read our directions. "Write the numeric expression in simplest radical form." Sounds crazy, I know, but what that's asking me to do is to rewrite this problem as a product of factors, okay? I'm gonna start out by doing this. I want to figure out factors of 50, and I want one of those factors to be a perfect square, okay? I've got my handy-dandy list of the commonly used perfect squares. Just start running them down. Can you factor 50-- would any of these numbers work, where any of these numbers times something else is 50? Let's look. Mm-hmm. Twenty-five. Sometimes there's more than one answer. There may be more than one way I could get 50 by using one of these numbers, but pick the largest number. In this case, I know I can multiply 25 times 2 and get 50. So, I'm gonna go back to the problem and rewrite that as 25 x 2.

(Describer) ...under a radical.

I essentially replace 50 and wrote it by its factors instead. And I know that that 25 is the factor that's a perfect square. Now that I have it written like this, I'm gonna break this up and rewrite it as the square root of 25 times the square root of 2. Didn't change the meaning. Math says, "That's fine. You can make that jump." It started out with both numbers under the radical, and I separated them. Now I want to take the square root of the number that is the perfect square. So, 25. I know that is a perfect square. What's the square root of 25? What number times itself is 25? Five. I'm gonna rewrite that as 5. Two's not a perfect square. I'm not gonna take it. You could put it in your calculator, and you'll get a long decimal that never ends and never repeats. Your calculator will stop it at one point, but in real life, it keeps going. I don't want to take the square root of 2. I don't want to write that long decimal. I'm keeping it as the square root of 2. So, the square root of 50 simplifies to 5 square roots of 2. And that is my answer. Okay? This is one of those problems we have to do a few times to get the hang of it.

(Describer) She changes the screen.

"Write the numeric expression in simplest radical form." They want me to simplify the square root of 12. What do we do first? First, we want to rewrite 12 as a product of its factors. I want one of those factors to be a perfect square. I'm looking in my list of the ones we use most often in Algebra I. Is there any number in this list that I can use as a factor of 12? You could use 4 because 4 times 3, that's 12. I'm gonna rewrite that 12. Got my square root sign. Gonna rewrite 12 as 4 times 3. Notice I always write the perfect square factor first because it helps me get through. After I've rewritten this as the square root of 4 times 3, I'm gonna separate it. I'm gonna write, it's the same as the square root of 4 times the square root of 3. That first number, because of how we're writing it, you always take the square root of it. What is the square root of 4? It's 2. Two square roots of 3. That is my answer for this one. Some people also say this, "2 root 3." "Two square roots of 3." "Two times the square root of 3." There are different ways to say it, but it means this: 2 root 3. Look at another one. All right. "Write the numeric expression in simplest radical form." This time we have the square root of 20. Let's search this list. Is there any number in here that's a factor of 20? If there's more than one, pick the biggest one. Did you spot it? It's 4 again-- 4 times 5. That'll give you 20. So, I'm gonna rewrite this as the square root of 4 times 5. Now that I've represented it like that, I'm gonna chunk it, separate it. I'm gonna write it as the square root of 4 times the square root of 5. What is the square root of 4? It's 2.

(Describer) She writes that.

The square root of 5-- 5 isn't a perfect square. I don't want to take it because I don't want the decimal. My answer for this one, 2 times the square root of 5. Do you see the process I'm taking every time? I've got my problem. I find which one of these perfect squares would work as a factor. I rewrite the problem using that perfect square factor. I chunk it up. I simplify. Let's try another one. This time, they want me to simplify the square root of 32. Let's look at our list. We're thinking or using the calculator, whichever you want to do. Did you find it? It's 16. 16 times 2, that is 32. What am I gonna do? Rewrite that square root of 32... as 16 times 2. Now that I know my factors, I'm gonna chunk it up. The square root of 16 times the square root of 2. What is the square root of 16? You're right. It's 4. So, 4 square roots of 2. And you're all done. Okay? All right. Go and try a few. Press pause. Take a few minutes. Work through these. Take your time. Don't press play until you're ready. When you want to see what I got, press play.

(Describer) Title: Write the numeric expression in simplest radical form. Number One: the square root of 500. Number Two: the square root of 18. Number Three: the square root of 75. Number Four: the square root of 48.

(female describer) Write the numeric expression in simplest radical form. Number 1: the square root of 500. Number 2: the square root of 18. Number 3: the square root of 75. Number 4: the square root of 48. Let's see how your answers compare with mine. So, square root of 500. That simplifies... to 10 square roots of 5. Square root of 18. That's 3 square roots of 2. Square root of 75. That's 5 square roots of 3. Square root of 48: 4 square roots of 3. Okay? Let me show you how I got those.

(Describer) She switches the screen to the first problem.

Okay. The square root of 500. Those two zeros are a hint that one of the factors is 100, another number with two zeros on the end. I know I can rewrite the square root of 500... Need my pen, or I'm not gonna be writing much of anything. I can rewrite that as 100 times 5.

(Describer) ...under a radical.

Remember, I always write the factor that's the perfect square first to help me move all the way through this problem. Now that I know these are the factors, I'm gonna chunk them up. So, the square root of 100 times the square root of 5. What is the square root of 100? Ten. Very good. So, 10 square roots of 5. And you're all done. That's how I got that first one. Remember the process. We write that number under the radical by its factors, chunk it up, simplify. All right, let's see number two. Here I needed to simplify the square root of 18. Look at our list. And it's 9. That's the perfect square factor for 18. I'm gonna rewrite the square root of 18... as the square root of 9 times 2, because 9 times 2 that's gonna give me 18. Now that I know the factors, chunk it. So, the square root of 9 times the square root of 2. What is the square root of 9? Good. It's 3. So, 3 square roots of 2. That's how I got that second one. You notice when I write my final answer, I don't write that multiplication dot? You don't see it when you see your answers in Algebra I. You know when you see two terms next to each other, you're multiplying them together. It's not wrong to write it. The third one. Whenever I see numbers like 25, 50, 75, 100, I always think quarters. Because in these kinds of problems, 25 is basically gonna be the factor you're looking for. I know that 75 is gonna factor to 25 times 3. So, I'm gonna rewrite that as 25 times 3. I rewrote it. Now it's time to chunk it. The square root of 25 times the square root of 3. And what is the square root of 25? It's 5. I just bring down that square root of 3. That's how I got that third one, 5 root 3. All done with that one. Let's look at how I got that last one. Square root of 48. So, you look at your list. This one I gave you kind of a thinker. You may have considered that 4 times 12 is 48, and you're not wrong. 4 times 12 is 48. But there's a bigger number in this list that's also a factor of 48. It's 16. That's actually the biggest factor. 16 times 3 is gonna give me 48. So, I'm gonna rewrite this as the square root of 16 times 3. Okay? Now I'm gonna chunk it. So, the square root of 16 times the square root of 3. And what is the square root of 16? It's 4. So, 4 root 3. And that's how I got that last one. Okay. All right, guys. Hope you're feeling comfortable about how to simplify radicals involving whole numbers and square roots. Hope to see you back here soon. Bye.

(Describer) Accessibility provided by the U.S. Department of Education.

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In this program, students learn the basics of simplifying square roots by factoring the whole number. Part of the "Welcome to Algebra I" series.

Media Details

Runtime: 17 minutes

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