Explainer: Approximating Radicals

In this explainer, we will learn how to approximate the nth root of a number and then use it to solve real-world problems.

A perfect square is the square root of a square number; for example, 9=3. The square root of a fraction made of perfect squares is a fraction; for instance, 1625=45. But the square root of a number that is neither a perfect square nor a fraction made of perfect squares is an irrational number.

Definition: Irrational Numbers

An irrational number is a real number that cannot be expressed as a simple fraction (i.e., whose numerator and denominator are integers).

Recall, when seeking to find the square root of a number, that it is equivalent to finding the length of the side of the square whose area is equal to this number. In this explainer, we will develop a method to estimate the value of an irrational square root.

Let us take 3, for example. A square with side length 3 has an area of 3. To estimate the square root, we find the square of closest size, whose area is a perfect square. The number 3 is between the square numbers 1 and 4. As 3 is closer to 4 than 1, the closest square in size to the square of area 3 is the square of area 4 and side 2.

Therefore, the nearest integer to 3 is 2.

How to Estimate the Value of the Square Root of an Irrational Square Root to the Nearest Integer

Let us consider a square of area 𝑎. Its side is 𝑎. To estimate the value of 𝑎, we need to find the nearest square number to 𝑎, which we call SN.

SN is either larger or smaller than 𝑎.

The value of 𝑎 can then be estimated as SN.

Remember that since SN is a square number, its square root is an integer.

In the first example, we will look at how we can identify the relative positions of square roots on a number line.

Example 1: Placing Square Roots on a Number Line

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

Answer

The numbers 𝑎, 𝑏, and 𝑐 are the sides of three squares. We have 𝑎<𝑏<𝑐; therefore, the corresponding areas are in the same order: 𝑎2<𝑏2<𝑐2. Since 102<111<120, we deduce that 𝑎=102.

Example 2: Finding Which Irrational Square Root Is Closest to Five

Which of the following numbers is closest to 5?

  1. 8
  2. 94
  3. 24
  4. 120
  5. 106

Answer

All the proposed answers are in the form 𝑎. Therefore, we are looking for the square root of number 𝑎 so that 𝑎 is close to 5. If we imagine two squares of sides 𝑎 and 5, their areas are 𝑎 and 25. Among the proposed numbers in the form 𝑎, we need to identify where 𝑎 is closest to 25.

The answer is thus 24 (answer C).

Example 3: Estimating the Value of an Irrational Square Root to the Nearest Integer

The formula for the area of a square is 𝐴=𝑠2, where 𝑠 is the side length. Estimate the side length of a square whose area is 74 square inches.

Answer

We need here to estimate 74. We therefore need to find the closest square number to 74. Considering 82=64 and 92=81, we find that the square number closest to 74 is 81 since 8174<7464. The nearest integer to 74 is thus 9.

Our estimate of the length of the square of area 74 square inches is 9 inches.

Let us now consider the two consecutive square numbers, SN1 and SN2, between which a number 𝑎 lies. We have SN1<𝑎<SN2, and thus SN1<𝑎<SN2.

It means that we can always find the two consecutive integers (SN1 and SN2 are necessarily consecutive integers) that a square root lies between.

Example 4: Finding the Irrational Number That Lies between Two and Three

Which of the following is an irrational number that lies between 2 and 3?

  1. 52
  2. 3
  3. 7
  4. 10
  5. 2.6

Answer

We are looking for an irrational square root that lies between 2 and 3, that is, the square root of number 𝑎 so that 2<𝑎<3. If we think of these three numbers as being the lengths of the sides of three squares, we can write about their respective areas that 4<𝑎<9. This is of course equivalent to simply squaring the three numbers.

Among the proposed answers, only answers B, C, and D are irrational numbers. And B is the only option whose square is between 4 and 9.

The answer is that 7 lies between 2 and 3 (answer C).

So far, we have found how to find the two consecutive integers between which a square root lies. We are going to see how we can find more accurate estimates. Let us consider, for instance, 5. We have 4<5<9, so 2<5<3.

Remember that 5 and 5 represent the side and the area of a square.

We can find the areas of the squares with sides 2.1, 2.2, 2.3, and so on, up to 3.

We see that 4.89<5<5.29, which means that 2.2<5<2.3.

We can apply exactly the same reasoning to be able to frame 5 between two consecutive two-decimal-place numbers. However, we can avoid working out all the areas of the squares of sides 2.21, 2.22, 2.23, and so on, up to 2.3. Let us see how. The idea is to work out the area of one square, possibly close to where we believe the square of side 5 lies. Since 5 is closer to 4.89 than 5.29, we can expect 5 to be closer to 2.2 than to 2.3. So, we can work out, for instance, the area of the square of side 2.23, that is, 2.232. We find 4.9729.

This value of 4.9729 is less than 5, which means that 2.23 is less than 5. Let us work out 2.242; it gives 5.0176.

Since 4.9729<5<5.0176, we have 2.23<5<2.24.

Here, we go from one inequality (about areas) to the other (about squares’ sides) by taking the square root of the three numbers. We can of course apply the same reasoning again if we want to find between which two consecutive three-decimal-place numbers 5 lies, and so forth.

Let us now apply this method in the following example.

Example 5: Finding between Which Two Consecutive One-Decimal-Place Numbers a Square Root Lies

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

Answer

We want to find between which two consecutive one-decimal-place numbers 151 lies. Remember that 151 is the side of the square of area 151. We need to find first the two closest square numbers between which 151 lies. We find that 144<151<169, which can be interpreted as the area of the square of side 151 that lies between the areas of the squares of sides 144=12 and 169=13.

Therefore, we have 12<151<13.

Now, to find between which two consecutive one-decimal-place numbers 151 lies, we are going to work out the area of a square whose side is a one-decimal-place number between 12 and 13. As 151 is closer to 144 than 169, we can pick a one-decimal-place number closer to 12 than to 13, for instance, 12.4. We find that 12.42=153.76.

This value is larger than 151. Therefore, we need to check the area of the square of side 12.3. We have 12.32=151.29.

We see that this area is closer to 151 but still higher than 151. Let us work out the area of the square of side 12.2; this is 12.22=148.84.

Now, we can write that 148.84<151<151.29, and therefore 12.2<151<12.3.

Key Points

  1. The square root of a square number, also called a perfect square, is an integer; for instance, 9=3. The square root of a fraction made of square numbers is a fraction, for instance, 1625=45. But the square root of a number that is neither a square number nor a fraction made of square numbers is an irrational number.
  2. An irrational number is a real number that cannot be expressed as a simple fraction (i.e., whose numerator and denominator are integers).
  3. The value of an irrational square root 𝑎 can be approximated first by finding the two nearest square numbers to 𝑎, SN1 and SN2. We have SN1<𝑎<SN2, and thus SN1<𝑎<SN2. We can approximate the value of 𝑎 more accurately by finding between which two consecutive one-decimal-place numbers 𝑎 lies. For this, we consider the 9 squares whose side lengths are comprised between the two consecutive integers SN1 and SN2, where each side is 0.1 larger than the side of the previous square. The area of each square can be calculated by squaring its side. Then, we identify between which two square areas 𝑎 lies, from which we deduce that 𝑎 lies between their two side lengths.
  4. For nonnegative numbers 𝑎, 𝑏, and 𝑐, if 𝑎<𝑏<𝑐, we can conclude that 𝑎<𝑏<𝑐. Furthermore, we can go from the bottom expression to the top. Hence, we can either take the square of or the square root of the three numbers or expressions in the inequality.

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