Calculating square roots is not new to mathematics students. However, one likely forgets the fundamentals in the course of progressing and learning the advanced ones. If that’s the case there is nothing you should worry about as through this article not only all your fundamentals will be revised but also you will take away interesting learning. Through this article, we will learn about irrational square roots of 2 and 6 using properties. We will begin by learning the meaning, and types and then go forward to learn the properties and finally learn about the interesting observation. Therefore, make sure that you go through the entire article thoroughly.

Table of Contents

**What Is Square Root & What Are Its Types?**

The terms squares and square roots can be considered particular exponents. The square root of any number can be understood as the factor of that number which, on multiplying by itself, gives the original number. The perfect squares are those positive numbers that can be expressed as the product of a number by itself. To calculate the square root of any number we can bring four methods into use. The names of the four methods are mentioned below.

- Estimation Method
- Long Division Method
- Prime Factorization Method
- Repeated Subtraction Method

Here, one must understand that the first three methods can be suitably used for numbers that are perfect squares. While the fourth method, which is the long division method, can be used for any number whether it is a perfect square or not.

**What Are The Properties Of Square Roots?**

Go through the points below and know the properties of square roots. Make sure that you give all the points a thorough read in order to understand the properties well.

- Firstly, if a number ends with an odd number of zeros, then it cannot have a square root. A square root is only possible for an even number of zeros.
- A perfect square root exists for a perfect square number only and it cannot be negative hence the square root of a negative number is not defined.
- Numbers that end with 1, 4, 5, 6, or 9 will have a square root while if the unit digit of a number is 2, 3, 7, or 8 then a perfect square root is not possible.
- The square root of an even perfect square is even and an odd perfect square will have an odd square root.
- Two square roots can be multiplied. √5, when multiplied by √2, gives √10 as a result.
- Two same square roots are multiplied to give a non-square root number. When √25 is multiplied by √25 we get 25 as a result.

**How To Prove That The Square Roots Of 6 & 2 Are Irrational?**

- We can prove that square root 6 is irrational by observing that with the value of √6, we get a decimal number that does not terminate and terms are not repeating themselves after the decimal point.
- Thus, the value obtained for the root of 6 satisfies the condition of being a non-terminating and non-repeating decimal number that keeps extending further after the decimal point which makes √6 an irrational number. Hence, √6 is an irrational number.
- Similarly, you can prove that square root 2 is irrational. √2 gives a decimal number and it keeps extending. Since it does not terminate or repeat after the decimal point, √2 is an irrational number.

We hope this article helped you with the basics and also successfully helped you realize an interesting observation. For more such informative articles stay connected with us.