Video: Estimating Radicals Using Number Lines

In this video, we will learn how to estimate the 𝑛th root of a numbe using number lines.

17:04

Video Transcript

In this video, we will learn how to estimate, how to find, the 𝑛th root of a number using number lines. This is also known as estimating radicals or surds. Let’s begin by defining what we mean by radicals.

A radical is an expression containing a radical symbol. On its own, the radical symbol represents square root. If we have a three as a superscript, this means the cube root. With a four, it’s called the fourth root. And the general format with the variable 𝑛 is called the 𝑛th root. To see what these roots mean, let’s look at a few examples.

We have the square root of nine, the cube root of eight, and the fourth root of 10,000. The square root of nine is asking, what number multiplied by itself gives us an answer of nine? We know that three times three is equal to nine. Therefore, the square root of nine is three. Square rooting is the opposite or inverse of squaring.

When it comes to cube rooting, we ask ourselves what number multiplied by itself multiplied by itself again gives us the answer. In this case, 𝑥 multiplied by 𝑥 multiplied by 𝑥 is equal to eight. Two cubed is equal to eight. Therefore, the cube root of eight is two. We repeat this process to calculate the fourth root of 10,000. 10 multiplied by 10 is 100. Multiplying this by 10 gives us 1,000 and multiplying by 10 again gives us 10,000. 10 to the power of four is 10,000. So, the fourth root of 10,000 is 10.

All three of these questions have integer solutions. This means that all three of the radicals, the square root of nine, the cube root of eight, and the fourth root of 10,000, are rational. They can be written as an integer or as a fraction. When we’re dealing with radicals, however, we often 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. Examples of irrational numbers are 𝜋 and the square root of three. If we type the square root of three into our calculator, we get an answer of 1.732050808 and so on. This is a decimal that does not terminate and does not recur. Therefore, it cannot be written as a fraction. In terms of this video, we will look at these irrational values and try and find approximate values for them.

If we consider two squares, the first with a side length of three and the second with a side length of four, we know that the area of the first square will be equal to three squared. And the area of the second square will be equal to four squared. Three squared is equal to nine, and four squared is equal to 16. Based on these diagrams, we can conclude that the square root of nine is equal to three and the square root of 16 is equal to four.

Let’s now assume that we had a third square where we didn’t know its side length, but we did know that its area is 12. If the area is equal to 12, the side length will be equal to the square root of 12. As 12 is greater than nine and less than 16, the square root of 12 has to be greater than three and less than four. This is a way of estimating radicals by finding the square number immediately above 12 and the square number immediately below 12.

This can also be shown on a number line. If we consider the number line between the integers three and four, as these values are equal to the square root of nine and the square root of 16, we know that the square root of 12 must lie between three and four. As 12 is slightly closer to nine than 16, we would expect the square root of 12 to be closer to three than four. It will be slightly below 3.5. We will now look at some specific examples using number lines.

Identify which of the arrows represents the position of the square root of 30.

In order to answer this question, we firstly need to recall our square numbers. The square numbers can be calculated by multiplying an integer by itself. The first square number is one as one squared or one multiplied by one is one. The second square number is four as two squared is equal to four. Three squared is equal to nine. The remaining square numbers up to 10 squared are 16, 25, 36, 49, 64, 81, and 100.

As square rooting is the inverse or opposite of squaring, we know that the square root of nine is equal to three. In the same way, the square root of 16 is equal to four. We can repeat this process matching up the radicals or surds with their integer values on the number line, as shown. In this question, we are asked to identify the arrow that represents the position of the square root of 30. As 30 lies between 25 and 36, the square root of 30 must be greater than the square root of 25 and less than the square root of 36. This means that it must lie between the integer values five and six. The correct answer is, therefore, arrow b.

In our next question, we need to work out which integer our radical is closer to.

Identify which arrow represents the position of the square root of 24 on the number line.

We recall that squaring and square rooting are inverse operations. Four squared is equal to 16 as four multiplied by four is 16. This means that the square root of 16 is equal to four. This means that the radical, the square root of 16, will be at the same place on the number line as four. Repeating this process for our other integer value five, we have five squared is equal to 25, which means that the square root of 25 is five.

In this question, we need to identify whether arrow a or arrow b represents the square root of 24. As 24 lies between 16 and 25, the square root of 24 must be greater than the square root of 16 and less than the square root of 25. This means the square root of 24 lies between the integers four and five. 24 is closer to 25 than 16. Therefore, the square root of 24 will lie closer to five than it does to four. The correct answer is, therefore, arrow b.

In our next question, we need to identify the correct statement.

Which of the following statements is true? All five of the statements want us to evaluate the square root of 17. Option (A) says, the answer is between three and 3.5. Option (B) the answer is between 3.5 and four. Option (C) the answer is between four and 4.5. Option (D) the answer is between 4.5 and five. Or option (E) the answer is between five and 5.5.

We will begin this question by drawing a number line to establish which two integer values the square root of 17 lies between. Let’s consider the integer values from three to six. We know that square rooting a number is the opposite of squaring it. As three squared is equal to nine, the square root of nine is equal to three. This means that we can put the radicals, the square root of nine, square root of 16, square root of 25, and square root of 36, on our number line.

The number 17 lies between 16 and 25. Therefore, the square root of 17 is greater than the square root of 16 and less than the square root of 25. We know that the square root of 16 is equal to four and the square root of 25 is equal to five. We can, therefore, conclude that the radical, square root of 17, lies between four and five. We can, therefore, eliminate options (A), (B), and (E).

4.5 is halfway between four and five. 17 is much closer to 16 than 25. It is one away from 16 but eight away from 25. This means that the square root of 17 will be closer to the square root of 16 than the square root of 25. We can, therefore, conclude that the square root of 17 lies between four and 4.5. This means that statement (C) is true. If you evaluate the square root of 17, the answer is between four and 4.5.

The last two questions we look at involve estimating cube roots.

Which of the following numbers is represented by the arrow on the number line? Is it (A) the cube root of 12, (B) the cube root of 15, (C) the cube root of 27, (D) the cube root of 45, or (E) the cube root of 64?

In order to answer this question, it is worth initially considering our cube numbers. In order to cube a number, we multiply it by itself and itself again. This means that one cubed is equal to one. Two cubed is equal to eight as two multiplied by two is four, and multiplying this by two gives us eight. Three cubed is equal to 27. Continuing this list for the integer values we have on our number line, we have 64, 125, 216, and 343.

Cube rooting is the opposite or inverse of cubing. Therefore, the cube root of eight is two. We can, therefore, match up the radicals, the cube root of one, cube root of eight, cube root of 27, and so on, with the integer values one to seven. The arrow on the number line lies between three and four. This means that our answer must be greater than the cube root of 27 and less than the cube root of 64. The only one of our five values that lies between these two is the cube root of 45. The correct answer is option (D). The cube root of 45 is greater than three and less than four.

Options (C) and (E) cannot be correct as they are equal to three and four, respectively. These are integer values; therefore, the cube root of 27 and the cube root of 64 is rational. The cube root of 12 and the cube root of 15 would both lie between two and three as 12 and 15 are greater than eight but less than 27.

Identify which of the arrows represents the position of the cube root of 23.

In order to answer this question, let’s firstly consider the cube numbers. In order to cube a number, we multiply the number by itself and then itself again. Therefore, one cubed is equal to one, two cubed is equal to eight, three cubed is equal to 27. This list would continue for four cubed, five cubed, and six cubed, as shown. We know that cube rooting is the inverse or opposite of cubing. Therefore, the cube root of eight is two, the cube root of 27 is three, the cube root of 64 is four, and so on.

We’ve been asked to identify the arrow that represents the cube root of 23. 23 lies between the two cubed numbers eight and 27. This means that the cube root of 23 is greater than the cube root of eight but less than the cube root of 27. These values are equal to two and three, respectively. The cube root of 23 lies between the integer values two and three. This means that the correct arrow is arrow b. The radical, the cube root of 23, lies between two and three.

We will now summarize the key points from this video. We can estimate the value of an irrational square or cube root by finding the nearest two square or cube numbers. For example, we know that 30 lies between the square numbers 25 and 36. This means that the square root of 30 must lie between the square root of 25 and the square root of 36. The square root of 25 is equal to five, and the square root of 36 is six. Therefore, the square root of 30 is greater than five and less than six. It lies between the two integer values five and six.

We also found that we can use a number line to identify which integer value a radical is closest to. As in the example above, we know that the square root of 26 lies between five and six. If we consider the number line between five and six, we know that the midpoint is 5.5. This is halfway between the square root of 25 and the square root of 36. As 26 is significantly closer to 25 than 36, the square root of 26 will be closer to the square root of 25. We can, therefore, conclude that the square root of 26 is closest to the integer value five.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.