[Maths Class Notes] on Square Root of 8 Pdf for Exam

The square root of 8 is 2√2 and its value in the decimal form is approximately equal to 2.8284. The square of a number, x, is obtained when it is multiplied by itself, i.e.x. The square root of the resulting number, x², is expressed as √x², which comes out to be x. 

Here, the square root of 8, denoted by √8, 8^0.5 or 8^1/2 is the number whose square gives you the number 8. Any square root is represented by the square root symbol √. This symbol is known as the radical symbol or simply radix. The number underneath this radical symbol is known as radicand. Hence, the number whose square root is to be determined is called the radicand. 

In this case, 8 is the radicand and 2√2 is the simplest and the radical form of √8. This value is called surd since you cannot simplify this further. 

Is Root 8 a Perfect Square?

Even though 8 is not a prime number, yet, when we take its square root, we get 2 as its only prime factor. Therefore, 8 is not a perfect square, hence, its value is represented in the root form.

However, 8 is a perfect cube of the number 2, since 2 x 2 x 2 = 8. So, you can easily find the cube root of 8 which is denoted as ³√8 = 2. 

However, there are a few numbers like 4, 9, 16, etc. that are perfect squares and whose squares are easy to find. 

√4 = 2, as you know 2 x 2 = 4

√9 = 3, as you know 3 x 3 = 9

√16 = 4, as you know 4 x 4 = 16

Is Root 8 Rational or Irrational?

We know that when we multiply an irrational number, with a rational number,the result obtained is an irrational number. Hence, the square root of 8, i.e. √8, is an irrational number 2√2. Also, the decimal form of √8 is a non-terminating decimal with non-repeating digits. Therefore, it can’t be written in the form of p/q, which again proves its irrationality. 

In this article, we will learn about the square root of 8 and how to find the root 8 value.

Value of Root 8

The value of root 8 is √8 = 2.82842712475. If you consider the first three decimal places, then √8 = 2.828

Now, since you know the value of root 8, let us find out how to calculate its value here. Although it is easy to find the root 8 value by using a calculator, you should know how to find its value manually.

Method to Find the Square Root of 8

There are two methods of estimating square roots, one that uses the prime factors of the number, and the other devises a division method. To estimate the square root of a perfect square, we always implement the first method. However, to calculate the square root of a number, which is not a perfect square, in decimal, the division method is used.

To find the value of √8, you have to first express the number 8 as a product of its prime factors. 

Now, 8 = 2³, that is 2 x 2 x 2. Therefore, 

√8 = √2³ or √2 x √2 x √2 

=> √8 = 2 x √2

=> √8 = 2√2

Now, we know that √2= 1.41421…

Therefore, √8 = 2 x 1.41421..

=> √8 = 2.82842..

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These are the steps to finding the value of the square root of 8 using the long division method:

Step 1: Pair the digits of the number starting from the right and put them under a bar.

Step 2: Think of a number whose square is less than or equal to 8. Here, the number is 2 as the square of 3 is 9, which exceeds 8. 

Step 3: Write 2 as the divisor and also as the quotient. We will get the remainder as 4.

Step 4: Now, double the divisor and write it on the left side. Here, it will be 4. 

Step 5: Now, find the largest possible digit to write on the right side of 4 that will also become the second digit in the quotient. The digit should be such that when multiplied by the both digits of the divisor, the resultant product is less than or equal to the dividend. Write the remainder and repeat the process to get the decimal places you want.

How to Estimate the Square Root of 8?

A faster way to estimate the square root of a bigger number is to determine its range. For example, you are calculating the square root of x. Consider any two other numbers, say, n and m in such a way that n² < x ². Now, you can say that √x is a number that lies in between the numbers n and m. In this way, it will become easier to estimate the square root of bigger numbers.

Now, applying the estimation method to find root 8, we know that 2²= 4, and 3²= 9. Hence the answer to the root of 8 lies between the numbers 2 and 3. However, since the square of 3 equals 9 which is larger than 8, the root 8 value lies in between the numbers 2.8 and 2.9.

The precise answer of the square root of 8 is 2.82842712475. This is much closer to the answer that we estimated.

The Square Root of Numbers from 1 to 100

Here is a chart of the square roots of numbers from 1 to 100 for your reference. You can use these values for your calculations.