Square Root

In simple mathematical terms, the square of a number is a value that we acquire when the original number is multiplied by itself, whereas the square root of a number is the factor of that number which, when multiplied by itself, yields the actual number. For example- if x is the square root of y, then x*x=y, that is, x²=y, which helps us derive the formula-√y. We know, 9 is the perfect square of 3, which means 3²=9, therefore, √9=3. The square root of every number is always positive which makes us conclude the fact that the original number can either be positive or negative. For example, both -3 and +3 are square roots of 9, although we find the positive numbers are mostly used.

Cube Root

Similarly, the cube of a number is a value that we obtain when the original number is multiplied thrice by itself, whereas the cube root of that number is the factor of the number which, when multiplied thrice by itself, produces the original number. For example- if a is the cube root of b, then, a*a*a=b, that is, a³=b, and the formula is ³√=b. Let’s take numbers to understand the concept better- 27 is the perfect cube of 3, so, 3³=27, which means, ³√27=3. 

Properties of the Square Root

Some important properties of the square root are as follows:

• When the original number is a perfect square, the square root is also perfect.
• Two square root values can be multiplied by each other. For example, √2 multiplied by √4 results in √8.
• If two square roots are multiplied, the result is a radical number, which means it has to be a non-square root number. For example, √4 multiplied by √4 is 4.
• Square roots of negative numbers are not defined since perfect squares cannot be negative.
• The square root of an even perfect square is even, and the square root of an odd perfect square is odd.
• Numbers ending with 1, 4, 5, 6, or 9 (in unit digit) have a perfect square root.
• In the case of numbers ending with 2, 3, 7, or 8 (in unit digit) perfect square root does not exist.
• If a number ends with an odd number of zeros (0’s), then it cannot contain a square root, whereas if the number has an even number of zeros, it can have a square root.

Properties of the Cube Root

Let’s read about a few properties of a cube root as well:

• The cube root of all the odd numbers is itself an odd number. For example- ³√27= 3.
• The cube root of the even numbers is itself even. For example, ³√8=2.
• The cube root of negative integers yields a negative value.

 Methods to Find Out Square Root of a Number

There are different methods on how to find the square root of a number. Some of them are:

•  Repeated Subtraction Method of Square Root- In this method, we have to subtract consecutive odd numbers from the given number till it reaches 0. The number of times we have subtracted is the square root value of the number.
•  Square Root by Prime Factorization Method- This is such a method in which the identical prime factors are paired up and the product of one element from each of the pairs yields the square root of the given number. The Prime Factorization Method can be used to find out if the number is a perfect square or not.
• Square Root by Estimation Method- The Estimation Method is used for estimating and approximating the square root of the given number. One disadvantage of this process is that it is a long and time-consuming one. 
•  Square Root by Long Division Method- This method is used for dividing large numbers into smaller parts for convenience. By using this method, the exact square root of a number can be found. 

To learn more, visit the Cuemath website. A detailed explanation is given about square roots and cube roots.