Note for Square Root and Cube Root - SRCR by Placement Factory
- Square Root and Cube Root - SRCR
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Class 8 β Chapter 7: Square Roots & Cube Roots (Lecture Notes) SQUARE OF A NUMBER: The Square of a number is that number raised to the power 2. Examples: Square of 9 = 92 = 9 x 9 = 81 Square of 0.2 = (0.2)2 = (0.2) x (0.2) = 0.04 PERFECT SQUARE: A natural number is called a perfect square, if it is the square of some natural number. Example: We have 12 = 1, 22 = 4, 32 = 9 Some Properties of Squares of Numbers 1. The square of an even number is always an even number. Example: 2 is even and 22 = 4, which is even. 2. The square of an odd number is always an odd number. Example: 3 is odd and 32 = 9, which is odd. 3. The square of a proper fraction is a proper fraction less than the given fraction. 1 1 1 Example: πππ’πππ ππ (2) = (2) Γ (2) = 1 4 1 and we see that 4 < 1 2 4. The square of a decimal fraction less than 1 is smaller than the given decimal. Example: 0.1 < 1 and (0.1)2 = 0.1 x 0.1 = 0.01 < 0.1. 5. A number ending in 2, 3, 7 or 8 is never a perfect square. Example: The numbers 72, 243, 567 and 1098 end in 2, 3, 7 and 8 respectively. So, none of them is a perfect square. 6. A number ending in an odd number of zeros is never a perfect square. Examples: The numbers 690, 87000 and 4900000 end in one zero, three zeros and five zeros respectively. So, none of them is a perfect square. SQUARE ROOT: The square root of a number x is that number which when multiplied by itself gives x as the product. We denote the square root of a number x by βπ₯. Example: Since 7 x 7 = 49, so β49 = 7, i.e., the square root of 49 is 7. 1
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METHODS OF FINDING THE SQUARE ROOTS OF NUMBERS To Find the Square Root of a Given Perfect Square Number Using Prime Factorization Method: 1. Resolve the given number into prime factors 2. Make pairs of similar factors 3. The product of prime factors, chosen one out of every pair, gives the square root of the given number. Examples: Find Square root of i) 625 and ii) 1296 5 5 5 5 2 2 2 2 3 3 3 3 625 125 25 5 1 625 = 5 x 5 x 5 x 5 Hence β625 = 5 Γ 5 = 25 1296 648 324 162 81 27 9 3 1 1296 = 2 Γ 2 Γ 2 Γ 2 Γ 3 Γ 3 Γ 3 Γ 3 Hence β1296 = 2 Γ 2 Γ 3 Γ 3 = 36 Test for a number to be a Perfect Square: A given number is a perfect square, if it can be expressed as the product of pairs of equal factors. Example: 1296 = 2 Γ 2 Γ 2 Γ 2 Γ 3 Γ 3 Γ 3 Γ 3 Hence β1296 = 2 Γ 2 Γ 3 Γ 3 = 36 To Find the Square Root of a given number By Division Method 1. Mark off the digits in pairs starting with the unit digit. Each pair and remaining one digit (if any) is called a period. 2. Think of the largest number whose square is equal to or just less than the first period. Take this number as the divisor as well as quotient. 3. Subtract the product of divisor and quotient from first period and bring down the next period to the right of the remainder. This becomes the new dividend. 4. Now, new divisor is obtained by taking twice the quotient and annexing with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of new divisor and this digit is equal to or just less than the new dividend. Repeat steps 2, 3 and 4 till all the periods have been taken up. Now, the quotient so obtained is the required square root of the given number. 2
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Example: Find the square root of 467856 6 46 36 10 10 128 1364 78 56 78 24 54 45 56 56 684 SQUARE ROOT OF NUMBERS IN DECIMAL FORM Method: Make the number of decimal places even, by affixing a zero, if necessary. Now, mark periods (starting from the right most digit) and find out the square root by the long-division method. Put the decimal point in the square root as soon as the integral part is exhausted. Example: Find the square root of 204.089796 1 24 2 1 1 282 2848 04 04 96 8 5 2 2 28566 .08 08 64 44 27 17 17 97 97 84 13 13 x 96 96 96 βJ204.089796 = 14.286 Square root of numbers which are not perfect squares Example: Find the value of β0.56423 up to 3 places of decimal. 7 145 1501 0.56 49 7 7 42 30 42 25 17 15 30 01 3 0.751 14.286
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2 29 β0.56423 = β0.564230 = 0.751 SQUARE ROOTS OF FRACTIONS: For any positive real numbers a and b, we have: i. βππ = βπ Γ βπ Example: Find the square root of ii. π βπ = β441 β1849 = 21 43 βπ βπ 9 Example: β 16 = β9 β16 = 3 4 CUBE OF A NUMBER: The cube of a number is that number raised to the power 3. Example: Cube of 2 = 23 = 2Γ2Γ2 = 8 PERFECT CUBE: A natural number is said to be a perfect cube, if it is the cube of some natural number. Example: 13 = 1 , 23 = 8, 33 = 27 and so onβ¦ CUBE ROOT: The cube root of a number x is that number which when multiplied by itself three 3 times gives x as the product. We denote the cube root of a number x by βπ₯ 3 Example: Since 5 x 5 x 5 = 125, therefore β125 = 5 METHOD OF FINDING THE CUBE ROOT OF NUMBERS: Cube Root of a Given Number by Prime Factorization Method 1. Resolve the given number into prime factors. 2. Make groups in triplets of similar factors. 3. The product of prime factors, chosen one out of ever triplet, gives the cube root of the given number. Example: Find Cube of 17576. 17576 = 2 x 2 x 2 x 13 x 13 x 13 3 Therefore β17576 = 2 Γ 13 = 26 Test for a Number to be a Perfect Cube A given natural number is a perfect cube if it can be expressed as the product of triplets of equal factors. Cube Roots of Fractions and Decimals 4
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