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Square and Square Root: Properties, Tricks, Examples

Square and Square Root

Square and square root:

Square number or perfect square:

 A natural number m is said to be square number or perfect square if it can be expressed In terms of n2, where n is also a natural number. It is necessary to know about square and square root to solve number system related problems.
Example: 4 is a square number as 4 = 22

Properties of perfect square:

  1. All perfect square numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place.
  2. None of these end with 2, 3, 7 or 8 at unit’s place.
  3. If a number has 1 or 9 in the unit’s place, then it’s square ends in 1.
  4. when a perfect square number ends in 6, the number whose square it is will have either 4 or 6 in unit’s place.
  5. There are 2n non-perfect square numbers between n2 and (n + 1)2
    Example:
    between 22 and 32 the no. of non-perfect square no. are 2×2 =4 as (5, 6, 7, 8)
  6. sum of first n odd natural numbers is n2 also Remember it for square and square root related terms.
    Example: 1+3+5 = 9 = 32
  7. Any perfect square number can be represented as sum of odd natural numbers starting from 1.
  8. Also, Any odd perfect square number can be represented as sum of two consecutive natural number.
    Example: 49 = 24+25, 121 = 60 + 61
  9. For any natural number (n + 1) × (n – 1) = n2 – 1.
    Example: 6 x 4 = 52 – 1 = 24
  10. Square numbers can only have even number of 0’s at the end.
  11. If a perfect square is of n-digits, then its square root will have \(\frac{n}{2}\) digits if n is even, or \(\frac{n + 1}{2}\) if n is odd?

Pythagorean triplet:

For any natural number a > 1, we have (2a)2+(a2 – 1)2=(a2 + 1)2

So, 2a, a2 – 1 and a2+ 1 forms a Pythagorean triplet.

By this rule, if any one of the members of Pythagorean triplet is given then you can find the rest. this also necessary for getting knowledge of square and square root.

Example:
1. 32 +  42 = 52 so, 3, 4, 5 are Pythagorean triplet.

  1. If 5 is one of the members of the Pythagorean triplet then find the rest.

Solution:
Try 2a = 5 then a will not be an integer so take next

a2 – 1 = 5 then a2 = 6 again not an integer
try a2 + 1 = 5 then a2 = 4 thus, a = 2

2a = 4
a2 – 1 = 3

Thus, Pythagorean triplet are 3, 4, 5.

Square root:

It is simply the inverse operation of square. Square and square root are in converse of each other.
There are many ways to find the square root of any number:

  1. By just inversion of square.
    Example: 32 = 9 and √9 = 3
  2. By repeated subtraction of odd numbers.
    Example: √9,
    9 – 1 = 8, 8 – 3 = 5, 5 – 5 = 0, by 3 times consecutive subtraction we got zero thus √9 = 3.
  3. By prime factorization.
    √36 = 2x2x3x3 = 22 x 32 thus square root of 36 is 2×3 i.e. 6
  4. By division method.
    Try to solve examples based on square and square root. So that it will be easy for you to solve problems related to it.
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