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Square and Square Roots


Square Number

if a natural number m can be expressed as n2, where n is also a natural
number, then m is a square number
Example
1=12
4=22
The numbers 1, 4, 9, 16 ... are square numbers. These numbers are also called perfect squares.

Important point to Note for Square Numbers

Square and Square Roots notes Class 8 maths CBSE

How to find the square of Number easily

Identity method

We know that
(a+b)2 =a2 +2ab+b2
Example
232= (20+3)2 =400+9+120=529

Special Case

(a5)2
= a(a + 1) hundred + 25
Example
252=2(3) hundred +25=625
 
Watch this tutorial for an example of square numbers

Pythagorean triplets

For any natural number m > 1, we have (2m)2 + (m2 – 1)2 = (m2 + 1)2
 So, 2m, m2 – 1 and m2 + 1 forms a Pythagorean triplet
Example
6,8,10
62 +82 =102
Lets take some solved example on it
Solved Example
Find a Pythagorean triplet in which one member is 12.
Solution:
If we take m2 – 1 = 12
Then, m2 = 12 + 1 = 13
Then the value of m will not be an integer.
So, we try to take m2 + 1 = 12. Again m2 = 11 will not give an integer value for m.
So, let us take 2m = 12
then m = 6
Thus, m2 – 1 = 36 – 1 = 35 and m2 + 1 = 36 + 1 = 37
Therefore, the required triplet is 12, 35, 37

Square Root

Square root of a number is the number whose square is given number
So we know that
m=n2
Square root of m
√m =n
Square root is denoted by expression √

How to Find Square root

Finding square root through repeated subtraction

We know sum of the first n odd natural numbers is n2. So in this method we subtract the odd number starting from 1 until we get the reminder as zero. The count of odd number will be the square root
Consider 36
Then,
(i) 36 – 1 = 35 (ii) 35 – 3 = 32 (iii) 32 – 5 = 27 (iv) 27 – 7 = 20
(v) 20 – 9 = 11 (vi)11 – 11 = 0
So 6 odd number, Square root is 6
 

Finding square root through prime factorization

This method, we find the prime factorization of the number.
We will get same prime number occurring in pair for perfect square number. Square root will be given by multiplication of prime factor occurring in pair
Consider
81
81= (3×3)×(3×3)
√81= 3×3=9
Example
 Find the square root of 57600.
Solution: Write 57600 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5×3×3
Therefore 57600= 2 × 2 × 2 × 2 × 5×3 = 240
 

Finding square root by division method

This can be well explained with the example
Step 1 Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar. So in the below example 6 and 25 will have separate bar
Step 2 Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend. Divide and get the remainder
In the below example 4 < 6, So taking 2 as divisor and quotient and dividing, we get 2 as reminder
Step 3 Bring down the number under the next bar to the right of the remainder.
In the below example we bring 25 down with the reminder, so the number is 225
Step 4 Double the quotient and enter it with a blank on its right.
In the below example, it will be 4
Step 5 Guess a largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
In this case 45 × 5 = 225 so we choose the new digit as 5. Get the remainder.
Step 6 Since the remainder is 0 and no digits are left in the given number, therefore the number on the top is square root
Square and Square Roots notes Class 8 maths CBSE
In case of Decimal Number, we count the bar on the integer part in the same manner as we did above, but for the decimal part, we start pairing the digit from first decimal part.
Example Find the square root of 870.25
Solution

Example
Find the Square root of 841
Solution
Square and Square Roots notes Class 8 maths CBSE

Estimating Digits in the Square Root

if a perfect square is of n-digits, then its square root will have n/2 digits if n is even or (n+ 1)/2
if n is odd
Or
We can create the pair in the number, the Number of pair gives the number of digit in the square root
Example
Find the number of digits in the square root of the following numbers.
(i) 25921
(ii) 37249
Solution
Here n is odd, so (n+1)/2 = 3
So three digits will be present in square root



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