Question

Define Square root. Properties of Square root. How to find Square root?

Answer 1

The square root of any number is equal to a number, which when squared gives the original number.

Let us say m is a positive integer, such that  √(m.m) = √(m²) = m

Square Root Symbol:

The square root symbol is usually denoted as ‘√’. It is called a radical symbol. To represent a number ‘x’ as a square root using this symbol can be written as:
‘ √x ‘

where x is the number. The number under the radical symbol is called the radicand. For example, the square root of 6 is also represented as radical of 6. Both represent the same value.

Square Root Formula:

The formula to find the square root is:

y = √a

Since, y.y = y² = a; where ‘a’ is the square of a number ‘y’.

Properties of Square Root:

• A perfect square root exists for a perfect square number only.
• The square root of an even perfect square is even.
• An odd perfect square will have an odd square root.
• A perfect square cannot be negative and hence the square root of a negative number is not defined.
• Numbers ending with (having unit’s digit) 1, 4, 5, 6, or 9 will have a square root.
• If the unit digit of a number is 2, 3, 7, or 8 then a perfect square root is not possible.
• 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.
• 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.

To find Square Root:

There are various methods for finding the square root of a number, such as estimation, the long division method. Here are some of the most common methods:

• Prime factorization method
• Repeated subtraction method
• Long division method
• Estimation method

(i) Square Root by Prime Factorization Method:

To find the square root of a given number through the prime factorization method, follow the steps given below:

Step 1: Factor the number into its prime factors.

Step 2: Write each prime factor twice in the expression for the square root, and simplify the expression as much as possible. 

Step 3: For each group of two identical prime factors, take the square root of one of the prime factors and place it outside the square root sign.

Step 4: Repeat the process for each group of prime factors.

Step 5: Find the product of the factors obtained by taking one factor from each pair.

Step 6: That product is the square root of the given number. 

(ii) Square Root by Repeated Subtraction Method:

Square of any perfect square number can be determined by the repeated subtraction method. In this method, we have to repeatedly subtract the given number by consecutive odd numbers until we get zero. The n^th odd number after which the subtraction yields zero, is the square root of the given number.

For example, the steps to find the square root of 81 by repeated subtraction method are as follows:

Step 1: 81−1 = 80

Step 2: 80−3 = 77

Step 3: 77−5 = 72

Step 4: 72−7 = 65

Step 5: 65−9 = 56

Step 6: 56−11= 45

Step 7: 45−13 = 32

Step 8: 32−15 = 17

Step 9: 17−17 = 0

Since, zero is obtained in the 9^th step, this means that the square root of 81 is 9.

(iii) Square Root by Long Division Method:

Long Division is a useful method for finding the square root of a non perfect square by hand. To find the square roots of imperfect squares such as 2,3,5,6,8 etc., we can use the long division method. Nonetheless, the long division method can be used to find the square roots of perfect squares too.

Let us understand this with the help of an example. For example, let us find the square root of 180 by long division method. 

Step 1: Place a bar over every pair of digits of the number starting from the units' place (right-most side), i.e., take the number in pairs from the right. 80 stands as a pair while 1 stands alone.

Step 2: We divide the left-most number by the largest number whose square is less than or equal to the number in the left-most pair, i.e., find a quotient which is the same as the divisor. 1×1 = 1. Multiply quotient and the divisor and subtract the result from 1.

Step 3: Bring down the number under the next bar to the right of the remainder. Add the last digit of the quotient to the divisor. To the right of the obtained sum, find a suitable number which, together with the result of the sum, forms a new divisor for the new dividend that is carried down, i.e., double the quotient obtained in step 2, which is, 1+1 = 2. The new divisor is now 2 × x. Bring down 80 as a pair and this will be the new dividend.

Step 4: The new number in the quotient will have the same number as selected in the divisor. The condition is the same as being either less than or equal to the dividend, i.e., find 2x × x such that the product is less than or equal to 80. We find 23×3 = 69. Subtract this from 80 and obtain the remainder. It is 11.

Step 5: Now, we will continue this process further using a decimal point and adding zeros in pairs to the remainder. 

Step 6: The quotient thus obtained will be the square root of the number. 

Thus √180 = 13.416, which is shown in the below image.

Therefore, the square root of 180 by long division method is 13.416.

(iv) Square Root by Estimation Method:

Estimation and approximation is a method to find the square root of imperfect square numbers, which refer to a reasonable guess of the actual value to make calculations easier and more realistic. This method helps in estimating and approximating the square root of a given number.

For example, let us find the square root of 15 by estimation method as follows:

Step 1: Find the nearest perfect square numbers to 15. 9 and 16 are the perfect square numbers nearest to 15.

Step 2: We know that √16 = 4 and √9 = 3. This implies that √15 lies between 3 and 4.

Step 3: Now, we need to see if √15 is closer to 3 or 4. Let us consider 3.5 and 4. Since 3.5² = 12.25 and 4² = 16. Thus, √15 lies between 3.5 and 4 and is closer to 4.

Step 4: Let us find the squares of 3.8 and 3.9.

Since 3.8² = 14.44 and 3.9² = 15.21. This implies that √15 lies between 3.8 and 3.9.

Step 5: We can repeat the process and check between 3.85 and 3.9. We can observe that √15 = 3.872.

Therefore, the square root of 15 by estimation method is 3.872.