squares and square roots

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Squares and square roots

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Since,4 can be expressed as 2*2=2 , 9 can be expressed as 3*3=3 ,all such no.s can be expressed as the product of the no. with itself. Such no.s like 1,4,9,16,25,… are known as square numbers. 2 3 Square numbers

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Number Square 1*1=1 2*2=4 3*3=9 4*4=16 5*5=25 6*6=36

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Perfect Squares The numbers 1,4,9,16… a re square numbers. These numbers are also called perfect squares.

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Properties of square numbers. The square numbers always ends with 0,1,4,5,6,9 .so,the no.s which ends with the above numbers are square numbers.

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Square roots Finding square roots with: Repeated subtraction method 2.Prime factorisation 3.Division method

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Repeated subtraction method e.g.,- √81 Sol.:- 81-1=80 (2) 80-3=77 (3) 77-5=72 (4) 72-7=65 (5) 65-9=56 (6) 56-11=45 (7) 45-13=32 (8)32-15=17 (9) 17-17=0 Result=9

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Prime factorisation e.g.,- √2352 2 2 2 2 3 7 2352 1176 588 294 147 49 7

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We have:2352=2*2*2*2*3*7*7 We have:2352*3=2*2*2*2*3*3*7*7 As,3 has no pair ,2352is not perfect square. So , we will multiply 2352*3 then :

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So, 2352*3=7056 √7056= 2*2*3*7=84

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Division method e.g., √529 Place a bar over every pair of digits starting from the digits at one’s place. Find the largest no. whose square is less or equal to the no. under the extreme left bar. take this no. as 2 529 -4 1

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t he divisor and the quotient with the no. under the extreme left bar as the dividend. Divide and get the remainder. 3. Bring down no. under the next bar to the right of the remainder . So the new dividend is 129. 4. Double the divisor and enter it with a blank on its 2 529 -4 129 2 529 -4 129 4_

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w hen the new divisor is multiplied to the new quotient the product is less than or equal to the dividend. 6. Since the remainder is o and no digits are left in the right . 5. Guess a largest possible digit to fill the blank which will also become the new 2 529 -4 129 4_

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given number, therefore √ 529 = 23 . 2 529 -4 129 43 129 - 0 3

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Square roots of decimals e.g., √17.64 1. To find the square root of a decimal we put bars on integral parts of the no. in usual manner. And place bars on decimal parts on every pair of digits

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Beginning with the first decimal place. Proceed as usual. We get 17.64 . 2.Now proceed in similar manner. The left most bar is on 17 and 4 <17<5. take this number as the divisor and the number under the left most bar as the dividend. Divide and get the remainder. 2 2 4 17.64 -16 1

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3. The remainder is1. write the number under the next bar to the right of this remainder, to get 164. 4.Double the divisor and enter it with a blank on its right. Since 64 is the decimal part so put a decimal point in the quotient . 5. We know 82*2= 164, therefore the new digit is 2. 4 17.64 -16 1 64 8_ 4. 17.64 -16 1 64 82

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Divide and get the remainder. 6. Since the remainder is 0 and no bar left, therefore √17.64 = 4.2. 4.2 17.64 -16 164 -164 0 82

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Made by:- Suhel Naryal 2126 8 th ‘B’

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