Lesson Video: Approximating Radicals Mathematics • 8th Grade

Video Transcript

In this video, we will learn how to approximate the 𝑛th root of a number. Another way to say that, we will learn how to approximate radicals.

First, we’ll start by defining what we mean by radicals. When we use the word radical, we’re talking about an expression containing the radical symbol. On its own, the radical symbol represents the square root of whatever is underneath the radical bar. If we have a three as a superscript with our radical, we call that the cube root. With a four, it’s called the fourth root. And then the general format if we use a variable 𝑛, it’s the 𝑛th root. To think about what these roots mean, let’s look at a few examples.

First of all, the square root of four is asking what number times itself equals four. We know that two times two equals four, and that makes the square root of four equal to two. When it comes to the cube root, there will be three of those values: 𝑥 times 𝑥 times 𝑥. In this case, if we want the cube root of eight, we’re saying, “what value when you multiply it together two other times equals eight?” We already know that two times two equals four times two would be eight, which makes the cube root of eight equal to two. In our final example, we need that value four times. I know that four times four equals 16 and that two times two equals four, which makes the fourth root of 16 equal to two.

All of the examples we looked at had integer solutions, which make these radicals rational. They can be written as an integer or as a fraction. But many times when we’re working with radicals, we will end up with irrational solutions. A real number that cannot be expressed as a fraction whose numerator and denominator are integers is an irrational number. We think of a number like 𝜋 but also something like the square root of three. If you type in the square root of three on your calculator, you’ll see that you get 1.732050808 continuing. It’s a decimal that does not terminate, and we know that this cannot be written as a fraction, which makes the square root of three irrational.

But many times, it’s useful for us to be able to take these irrational, radical values and find an approximate value for them. And that’s what we’re gonna look at now. If we have two squares, the first square has a side length of three, and the second square has a side length of four. This means that the area of the first square will be equal to three squared, the side length squared. And the area of the second square would be equal to four squared. Three squared is nine, four squared is 16. And based on this image, we can say that the square root of nine is three and that the square root of 16 equals four.

Now, let’s say that we had a third square and we didn’t know its side length, but we did know that it had an area of 12. If this square has an area of 12, then its side length will be equal to the square root of 12. The square root of 12 has to be greater than three and less than four. And this is a form of approximating radicals. We did this by finding the square number below 12 and the square number above 12, which told us the two integers that the square root of 12 will fall between.

Let’s consider one other example. This time, we have a third square whose area is 10, which means the side length we’re looking for will be the square root of 10. Now, it’s absolutely true by the same process that we can say that it’s going to fall between three and four. However, there is something else we can say about this. Because 10 is much closer to nine, we can say that while the square root of 10 will fall between three and four, its nearest integer will be three. That is to say, the square root of 10 is closer to three than it is to four. Using this information, we’re ready to look at some examples.

Michael is trying to find which two whole numbers lie on either side of the square root of 51. He decides that it will be helpful to use what he has been taught about square numbers. What is the greatest square number below 51? What is the smallest square number above 51? Hence, determine the two consecutive whole numbers that the square root of 51 lies between.

First of all, we should think about what we know about square numbers. A square number is the result of multiplying an integer by itself. For example, one squared is one, and two squared is four. So, we call four a square number. It’s helpful to memorize the first 10 or so square numbers. Three squared is nine. When we’re listing out square numbers, the first three are one, four, nine, then four squared, which is 16, five squared, which is 25, six squared, which is 36, seven squared, which is 49, eight squared, which is 64, then 81, and then 100. These are the first 10 square numbers, and we can use that to answer the questions below.

What is the greatest square number below 51? This means we’re looking for the square number that’s closest to 51 but not over. And that’s gonna be 49. The next question is similar. What is the smallest square number above 51? That means we need the square number that’s closest to 51 but must be greater than 51. And here, that’s going to be 64. If we put 49 and 64 on a number line, 51 would fall between them but closer to 49.

Now, on another number line, we could put the square root of 49 and the square root of 64. Again, when we add the square root of 51, it falls closer to the square root of 49 than it does to the square root of 64. But we know that the square root of 49 is seven, and the square root of 64 is eight. Which means we can say that the square root of 51 will fall somewhere between the consecutive whole numbers of seven and eight.

In our next example, we’ll again consider some perfect squares on a number line to help us solve for an irrational square root.

The positions of the numbers the square root of 120, the square root of 102, and the square root of 111 have been identified on the number line. By considering their size, decide which number is represented by 𝑎.

We’ve been given the square root of 120, the square root of 102, and the square root of 111. We don’t recognize any of these values as square numbers, which means the result will not be an integer. However, when we look on our number line, we do see two integer values; we see 10 and 11. And we know that 10 is equal to the square root of 100. 10 squared equals 100. And so, the square root of 100 equals 10. We can follow the same procedure to find what 11 would be. 11 is equal to the square root of 121. This is because 11 times 11 is 121. When we see the square root of 121, one of our values is the square root of 120. And the square root of 120 will be very close to the square root of 121 on a number line.

By that same reasoning, we know that the square root of 102 will be very near the square root of 100 on a number line. And we finally have the square root of 111, which falls about halfway between the square root of 100 and the square root of 121. Based on the data in the number line, we’re able to see that the square root of 102 would fall between 10 and 10.2. Our question was looking for the value that 𝑎 represented. And here, 𝑎 represents the square root of 102.

Let’s consider another example.

Which of the following numbers is closest to five? (A) The square root of eight, (B) the square root of 24, (C) the square root of 94, (D) the square root of 106, or (E) the square root of 120.

One way to solve this would be to think about the first 10 square numbers. One squared is one, two squared is four, three squared is nine, then 16, 25, 36, 49, 64, 81, and 100. And as we look at this list, we know that 25 is equal to five squared. If five squared equals 25, then the square root of 25 is five. Now, we are looking for the value that is closest to five. The exact value of five would be the square root of 25. And that means we are looking for the radical that is closest to the square root of 25. In our examples, we have the square root of 24, which is far closer to the square root of 25 than any of the other examples. The square root of 24, to the closest integer, would be five.

In our next example, we’ll think about how approximating radicals could help us when we’re working with area problems.

The formula for the area of a square is 𝐴 equals 𝑠 squared, where s is the side length. Estimate the side length of a square whose area is 74 square inches to the nearest integer.

We’re working with a square whose area is 74 square inches, and we want to know what the side length is equal to. Since the area equals the side squared, we also know that the side length equals the square root of the area. For our square, we can say that the side length will be equal to the square root of 74. But to approximate this value, we’ll want to find the square numbers above and below 74. You can either try and remember what would be above and below 74 or make a list of the first 10 square numbers.

From the list, we see that 64 is just below 74 and that 81 is just above 74. If 64 is the square number just below 74 and the square root of 64 is eight, the square number just above 74 is 81 and the square root of 81 is nine. At this point, we can confidently say that the square root of 74 will be between eight and nine. But in this case, we’re trying to round to the nearest integer. And that means we need to make the decision is the square root of 74 going to be closer to nine or to eight? And for that, we need to determine how far 74 is from 64, which is 10 units, and how far 74 is from 81, which is seven units.

Because 74 is slightly closer to 81, when we round to the nearest integer, we round to nine. However, we cannot forget the context of our question. This nine represents the side length of a square. And since the area was measured in square inches, the side length will be measured in inches. To the nearest integer, the side length of a square with an area of 74 inches squared is nine inches.

In our final example, we’re going to look at approximating a radical to the tenth place instead of to the nearest integer.

Find the two consecutive one-decimal-place numbers that the square root of 151 lies between.

This one decimal place is to the nearest tenth. And we want to look at the square root of 151. In solving problems like this, one strategy we often use is try and find a square number above and below the value we’re using. If we’re trying to find the square root of 151, we would want a square number both above and below this value. At first, you might think of 100, which is 10 squared. But when we think of 11 squared, that’s 121. And then 12 squared is 144. After that, we have 13 squared, which is 169. And that means we found the two values above and below 151.

When we go to put these values on a number line, we see that the square root of 151 is much closer to the square root of 144. This is because 151 is closer to 144 than it is to 169. We, of course, know that the square root of 144 is 12 and the square root of 169 is 13. And so, of course, we can say that the square root of 151 will be greater than 12 but less than 13. But sometimes, we want to be a bit more specific. To do that, imagine that we zoom in on this number line and we start with the square root of 144, which is 12. But then we want to look at the one-decimal-place numbers, which means we want 12.1, 12.2, and so on.

In order to decide where the square root of 151 will go, we calculate 12.1 squared, which is 146.41. If we jump over and look at 12.3 squared, we see that that is 151.29. It’s greater than 151. So, that means we found out that the square root of 151 cannot be more than 12.3. And that means we’ll need to check 12.2. 12.2 squared is 148.84, which is less than 151. And that means we found that the square root of 151 falls somewhere between 12.2 and 12.3. We can write this as the inequality 12.2 is less than the square root of 151 which is less than 12.3.

Before we finish, we just need to make a note about approximating higher roots. In all of our examples, we were approximating square roots. But cube roots and higher roots would follow the same procedure. For example, if we we’re dealing with cube numbers, you could list out the cube values. Make a number line. The cube root of one is one, the cube root of eight is two, the cube root of 27 is three. And that means if we were looking for the cube root of five, it would fall between one and two. And the same thing would work with the fourth root all the way up to the 𝑛th root.

We can summarize what we’ve learned with a few key points. The value of an irrational square root 𝑎 can be approximated by finding the two nearest square numbers to the square root of 𝑎. So that the square number one is less than 𝑎 and the square number two is greater than 𝑎. Therefore, the square root of square number one is less than the square root of 𝑎 which is less than the square root of square number two.