Square and Square Root ( as per PSEB )

Square and Square Root

Points to Remember:

1.   When a number is multiplied be itself then the product is the square of the number.

2.   Square of an odd number is odd.

3.   Square of an even number is even.

4.   Perfect square has only 0, 1, 4, 5, 6 or 9 digits at its unit’s place.

5.   If a number has its unit’s place digits 2, 3, 7 or 8 then it is not a perfect square.

6.   If a number has its unit’s place digits 1 0r 9 then its square has 1 as unit’s place digit.

7.    If unit’s place digit of a number is 2 or 8 then its square has 4 as unit’s place digit.

8.   If unit’s place digit of a number is 3 or 7 then its square has 9 as unit’s place digit.

9.   If unit’s place digit of a number is 4 or 6 then its square has unit’s place digit 6.

10. If unit’s place digit of a number is 5 then its square has unit’s place digit 5 and ten’s place digit 2 i.e. the square ends with 25.

11. If unit’s place digit of a number is 0 (zero) then its square has  

 unit’s and ten’s place digit as zero.

12. When a perfect square is divided by 3 remainder is 0 or 1.

Square and Square Root

Points to Remember:

1.   By multiplying a number by itself we get the square of the number.

2.   Square of positive and negative numbers is always a positive number and no negative number is a perfect square.

3.   By using the identities (a+b)² = a²+ 2ab + b² and (a – b) ² = a² – 2ab + b² we can find the square of numbers.

Points to Remember:

1.   If n= m2 then n is square root of m and we write Ön = m.

2.   If we go on subtracting 1, 3, 5, …… in order, at one stage the remainder will be zero. If the number of subtractions is n (say n times) then he square root of this number is n.

Points to Remember:

1.   Square root by prime-factorization method:

(i) Prime factors of number are made.

(ii) Pairs of equal prime factors of number are made.

(iii) One factor out of a pair of equal factors is taken.

(iv) By multiplying the one-one factor taken from pair of equal 

      factors square root of the number is obtained.

    2. If the prime factors are not in pairs then number is not a perfect   

        square because perfect square is formed by multiplying two equal

        numbers.

          We know that if equal prime factors are not in pairs then number is not a perfect square but this can be made another number by multiplying or dividing it by some prime factors which is a perfect square.

          To find the number of digits in the square root of a number, starting from unit place put dot (·) on the alternate digits of the number. The number of digits in square root is the same as the number of dots.

Points to Remember:

1. Rational number: A number which can be put in the form of p/q   

    Where p, q are integers, q ≠ 0 (i.e. q is not zero), p and q having  

    no common factor is called rational number.

2. For taking square root of rational number we write Öp = Öp

                                                                                    q    Öq         

Points to Remember:

1.   To find the square root of decimal numbers

(a) When these numbers are perfect squares:

 (i)  Starting from decimal on right and left side make the pairs.

(ii)  Start doing square root as usual.

(iii)  Whenever turn of decimals comes, put decimal in quotient.

(iv)  Proceed till remainder is zero. The obtained quotient is square root of the number.

        (b) When numbers are not perfect squares: In such questions to find out square root upto certain decimal places we put zero after the decimal point to complete the pairs because by putting zero after the decimal the value of number does not change.