Property 1: If the units digit of a number is 2,3,7 or 8, then it does not have root in N (the set of natural numbers). Example : 132, 433, 688 does not have perfect square roots as unit digits are 2,3,and 8 respectively.

Property 2: If a number ends in an odd number of zeros, then it does not have a square root. If a square number is followed by an even number of zeros, it has a square root in which the number of zeros in the end is half the number of zeros in the number. Example : 2000 does not have prefect square root as the number of zeroes are 3(odd). 900 have a perfect square root as number of zeroes are 2(even). So square root of 900 will contain only 1 zero.(half of two zeroes).√900 = 30.

Property 3: The square root of an even square number is even and that root of an odd square number is odd. Example : √144 = 12 (both are even numbers) and √225 = 15 (both are odd numbers).

Property 4: If a number has a square root in N, then its unit digit must be 0, 1, 4, 5, or 9. Example : Unit digit of √1024 is 2 as unit digit of 1024 is 4 and its square root is 2.

Property 5: Negative numbers have no square root in the system of rational numbers. Example : √(-9) is not a rational number. It will be complex number.

Property 6: The sum of first n odd numbers is n^{2}. Example :
1 + 3 + 5 = 9 = 3^{2} as there are 3 odd numbers so it will be 3^{2}.
1 + 3 + 5 + 7 + 9 = 5^{2} = 25 as there are 5 odd numbers.
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Practice on properties of Square Roots

Q.1 Write the possible unit digit of square root of :
1) 9801 2) 1156 3) 27225 (Answer)

Q.2 Find the sum of the following numbers without actually adding the numbers.
1) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15
2) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 (Answer)Squares and Square roots