Squares and Square roots Introduction of Squares and Square Roots: LECTURE - 1 If a number is multiplied by itsely, then the product is said to be the square of that number. i.e., If m and n are two natural numbers such that n m, then n is said to be the square of m. The square root of a number is that number which when multiplied to itsely gives the original number as the product. Let the square root of b is denoted by i.e., b a and b a Square Numbers: b A natural number is called a square or a square number if it is the square of any number. i.e., for a rational number m, if n m, then n is a square or square number e.g. 4, 16, 5, 81 etc. are square numbers. when a number is expressed as the product of its prime factors, then the expressed number is a square if its prime factors contain even powers only Note: No prime number can be a square number. Ex. 144 4 3 is a square number since its prime factors contain even powers only. it means 144 is a square of 1. Ex: 3 5 odd power even power is not a square number since one of its prime factor i.e. has odd power. Ex. 1: Determine whether 736 is a square number or not. Sol. Let us express 736 as a product of prime factors. 8 4 736 3 even powers Hence 736 is a square number. Properties of a square number: (i) Every square number ends with either, 1, 4, 5, 6, or 9. (ii) The number of zeroes at the end of a square of a number ending with zero, is always even. 1 1 1 1 Note: 49 is the only square number in which both the digits are square numbers. (iii) A square number can never end with the digits, 3, 7, or 8 ex. 4 3 7 not a square 63 3 7 not a square
(iv) The square of an even number is even and that of an odd number is odd. Ex. 1 1 11 11 (v) The square of a proper fraction is smaller than the fraction Ex. 3 4 9.44 and 3.66 3 < 3 (vi) For every natural number n, the sum of the first n odd natural numbers is n. Ex. 1 + 3 + 5 + 7 + 9 + 11 36 (6) sum of the first 6 odd numbers. (vii) Pythagorean Triplet: There positive numbers, m, n and p are siad to form a Pythagorean triplet, if m + n p, where p is the largest number. Ex. (3, 4, 5) is a Pythagorean triplet Since 3 + 4 9 + 16 5 (5) Note: For every natural number p > 1, we have (p) + (p 1) (p + 1) So (p, p 1, p + 1) forms a Pythagorean triplet. (viii)square of a number can never be negative i.e., it is always positive or zero. In other words, square of a negative as well as a positive number is always a positive number. Ex. ( 4) 16 (4) 16 (ix) If a number has 1, 5, 6 and in its unit place, then its square would also have 1, 5, 6 and in its unit place. Ex. 1 441 15 5 16 56 1 1 (x) There are n non-square numbers between the squares of the numbers n and (n + 1) Ex. 3 9 4 16 Here n 3, n + 1 4m b 6 between 9 and 16, we have 6 non square numbers (xi) The sum of square of first n natural numbers is 1 6 n(n + 1) (n + 1). Note: The square of any integer other than 1, and 1 is either a multiple of 4 or exceeds a multiple of 4 by 1. Squaring a number of the form 45 (where x is tens digit and 5 is units digit). (x5) (1x + 5) (1x + 5) (1x + 5) 1 x + 5x + 5x + 5 1x + 1x + 5 x(x + 1) 1 + 5 Ex. (45) 4 (4 + 1) 1 + 5 1 + 5 5
Square of a 3 digit number of the form 5 ab. If a number is of the form 5ab, where 5 is hundreds digit and a and b are its tens and units digit respectively, then (5 ab) (5 + ab) 1 + (ab) Ex. (51) (5 + 1) 1 + (1) 71 + 441 71441 Square a number of the form 5a (where a is unit and 5 is tens digit). (5a) (5 + a) 1 + a Ex. (57) (5 + 7) 1 + (7) 3 + 49 349
Squares and Square roots LECTURE - Square Roots: The square root of a number a is that number which when multiplied by itsely gives a as the result. i.e. a a a Note: The numbers whose square roots are whole numbers are called perfect squares. ex. 4, 9, 5, 81 etc. Square of a prime number is never a prime Square root of a negative number is not defined in the set of a real number. Methods of determining square roots: (i) By Repeated subtraction: Every square number can be expressed as the sum of successive odd numbers starting from 1. If a number is a perfect square and we subtract successive odd numbers from it, until we get zero as the final result. The number of stepin which we get zero is the square root of the number. Ex 1: Determine the square root of 169 by the method of repeated subtraction. Sol. 1. 169 1 168. 168 3 165 3. 165 5 16 4. 16 7 153 5. 153 9 144 6. 144 11 133 7. 133 13 1 8. 1 15 15 9. 15 17 88 1. 88 19 69 11. 69 1 48 1. 48 3 5 13. 5 5 we get in 13 th step so 169 13 (ii) By prime factorization: (a) Express the given number as a product of prime factors. (b) Make pairs of similar factors (c) The product of prime factors, after taking one factor out of every pair will give the square root of the number. Ex. Determine square root of 964. Sol. 964 7 4
(iii) By long division: 964 4 7 7 98 (a) Group the digits in pairs, starting with the digit in unit place. Each pair or remaining digit is called a period. (b) Take the largest number whose square is equal to or just less than the first period. Take this number as the divisor and also as the quotient. (c) Subtract the product of the divisor and quotient from the first period and bring down the next period to the right of the remainder. This becomes new dividend. (d) The new divisor is obtained by taking two times the quotient and enter it with a suitable digit which is also taken as the next digit of the quotient, taken in such a way that the product of the new divisor and this digit is just less than or equal to the new dividend. (e) Repeat these steps till all the periods have been taken up. Ex. Find square root of 13689. Sol. 117 1 13689 1 1 36 1 77 1589 13689 117 1589 Find square roots of (i) 169, (ii) 196, (iii) 964 Square roots of dicimal number: (a) Group the digit in pairs/periods from right to left in integral part and from left to right in the decimal part. (b) Put the decimal point in quotient when integral part is over and first group of the decimal is brought down. Ex. Find square root of 11.36. Sol. Square root of fraction: 1.6 1 11.36 1 1 6 136 136 11.36 1.6 If a and b are positive numbers, then square root of ab ab a b a square root of b a a b b Ex. Ecaluate 9 16 676 Sol. 9 16 676 1816 9 115 676 676 3 5 7 3 5 7 13 15 4 1 6 6 13
Squares and Square roots LECTURE - 3 Estimating a Square Root: In estimating a square root, we determine the two squares between which the given number lies. Ex. Estimate the square root of 15 Sol. Let us check squares of some numbers (1) 1, () 4, (3) 9 (4) 16 15 lies between 9 and 16 (35) 15 So, 15 lies between (35) and (4) (36) 196 (37) 1369 (38) 1444 (39) 151 15 lies between these two numbers i.e. 1444 < 15 < 151 (38) < 15 < (39) 15 is nearest to 151 15 39 (Approx.) Some problems based on finding the perfect squares (1) To find the smallest number by which given number is to be multiplied to make it a perfect square. (a) Factories the given number and form pairs of like prime factors. (b) Prime factors that are left unpaired are to be multiplied to given number to make it perfect square. Ex. Find the smallest number by which 7 must be multiplied to make it a perfect square. Sol. 7 3 3 3 5 1 3 1 () (3) (5) odd power to make it a perfect square, make all powers of prime factors as even Hence, 7 must be multiplied by 3 5 i.e., 3 to make it a perfect square. () To find the smallest number by which given number is to be divide to make it a perfect square. Ex. Find the smallest number by which 648 must be divided to make it a perfect square. Sol. 648 3 3 3 3 3 4 3 odd power even power
Hence 648 must be divided by () 1 i.e. to make it a perfect square. (3) To find smallest number to be added to the given number to make it a perfect square. (i) Find square of the given number by long division method. (ii) Square the successor of the quotient obtained. (iii) Subtract the given number from the obtained square. The result obtained after subtraction is the required number. Ex. Find the smallest number that should be added to 4571 to make it a perfect square. Sol. (i) By long division method: 4 41 1 4571 4 5 571 41 17 Square of successor of quotient () 484 Required number 484 4571 33 (ii) By estimation method: Given number is a 5 digit number, it could be square of 3-digit number. () 4, (3) 9 (1) 441, () 484 Q 4571 lies between 441 and 484. Hence required number 484 4571 33. (4) To find the smallest number to be subtracted from the given number to make it a perfect square. Ex. Find the smallest number to be subtracted from 5798 to make it a perfect square. Sol. (i) BY long division: 44 1 5798 4 179 176 48 38 38 Hence Remainder i.e. 38 should be subtracted from 5798 to make it a perfect square. required number 5798 38 576 (4) (ii) By Estimation method: Pick up the perfect square which is less than the given number Subtract the choosen perfect square from given number The difference obtained is the required number. (5) To find the the smallest number added/subtracted to make it a perfect square: Use the methods discussed above to solve such kind of problems. Ex. Find the smallest number to be added/subtracted from 6591 to make it a perfect square.