CONTENT: Squares and square roots. Properties of perfect square. Square root using factor method & division method for numbers containing not more than 4 digit ¬ more than 2 decimal places. Table of squares and square roots. Pattern of square numbers. Quick notes.

SQUARES:

SQUARES In mathematics square of a number is obtained by multiplying the number by itself. For example: square of “4” is 4 multiplied by 4,which is equal to “16”.

Perfect squares:

Perfect squares A Perfect square is a natural number which is the square of another natural number . For Example ; consider two number 84 and 36. The factors of 84 are 84 = 2X2X3X7 and 36 are 36 = 2X2X3X3. The Factor of 84 cannot be grouped into pairs of identical factors. So, 84 is not a perfect. But the factor of 36 can be grouped into pairs of identical factors , like 36 = 2X2 X 3X3 =6 2

PROPERTIES OF PERFECT SQUARES:

PROPERTIES OF PERFECT SQUARE S A number ending in 2,3,7or 8 is never a perfect square. A number ending in an odd number of zeros is never a perfect square. The square of even number is even. The square of odd number is odd. The square of a proper fraction is smaller than the fraction. The square of a natural number ‘n’ is equal to the sum of the first ‘n’ odd numbers . For example:n 2 is equal to the sum of the first ‘n’ odd numbers.

Square Roots:

Square Roots The square of number x is that number which when multiplied by itself gives x as a product. There are 2 methods to find square roots Prime factorization method Division method

PRIME FACTORIZATION METHOD:

PRIME FACTORIZATION METHOD In order to find the square root of a perfect square , resolve it into prime factors; make pairs of similar factors , and take the product of prime factors , choosing one out of every pair.

LONG-DIVISION METHOD:

LONG-DIVISION METHOD When numbers are very large , the method of finding their square roots by factorization becomes lengthy and difficult .So, we use long-division method.

SQUARE ROOTS OF NUMBERS IN DECIMAL FORM:

SQUARE ROOTS OF NUMBERS IN DECIMAL FORM For finding the square root of a decimal fraction , make the number of decimal places even by affixing a zero , if necessary; mark the periods , and find out the square root, putting the decimal point in the square root as soon as the integral part is exhausted.

Application of squares:

Application of squares Show that 6292 is not a perfect square? Solution: Resolving 6292 into prime factors, we get,6292=2*2*11*11*13=(2 2 *11 2 *13). Thus ,6292 cannot be expressed as a product of pairs of equal factors.Hence,6292is not a perfect square.

Application of squares Roots:

Application of squares Roots In an auditorium, the number of rows is equal to the number of chairs in each row. If the capacity of the auditorium is 2025,find the number of chairs in each row? Let the number of chairs in the rows be x. Then, the number of rows=x Total number of the chair in the auditorium=(x*x)=X 2 X 2 =2025 =5*5*3*3*3*3 x=(5*3*3)=45.

QUICK NOTES: If p=m 2 , where m is a natural number, then p is a perfect square. When the sum of odd numbers is even it is a perfect square of even number and when the sum of odd numbers is odd it is a perfect square of odd numbers. To find a square root of a decimal number correct up to “n” places , we find the square root up to (n+1) places and round it off to “n” places.

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