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Squares and Square Roots CLASS VIII CHAPTER 3 SQUARES AND SQUARE ROOTS Square number: If any natural number m can be expressed as n2, where n is also a natural number, then m is known as a square number. The square numbers are also called as perfect squares. Example: Let m = 36. Now, 36 can be expressed as 62, where 6 is a natural number. Therefore, 36 is a square number. Properties of Square Numbers: 1. The unit’s place of square numbers can be 0, 1, 4, 5, 6 or 9. No square number can end with 2, 3, 7 or 8. Exercise 3.1 Question 1. Which of the following natural numbers are perfect squares? Give reasons in support of your answer. (iii) 1024 (iv) 243 Solution: Question 2. Show that each of the following numbers is a perfect square. Also, find the number whose square is the given number. (iii) 3025 (iv) 3969 Solution: Question 4. Find the smallest natural number by which 5808 should be divided to make it a perfect square. Also, find the number whose square is the resulting number. Solution: Do the following questions Question 1. Which of the following natural numbers are perfect squares? Give reasons in support of your answer. (i) 729 (ii) 5488 Question 2. Show that each of the following numbers is a perfect square. Also, find the number whose square is the given number. (i) 1296 (ii) 1764 Question 3. Find the smallest natural number by which 1008 should be multiplied to make it a perfect square. If a number have 1 or 9 in its unit’s place, then square of that number will end with 1. 3. If a number have 4 or 6 in its unit’s place, then square of that number will end with 6. 4.There will always be even number of zeros at end of any square number. 5. On combining two consecutive triangular numbers we get a square number. Example: 6. There are 2n non-perfect square numbers between the squares of the numbers n and (n+1). Example: Between 32 = 9 and 42 = 16, there lies 6 numbers which are 10, 11, 12, 13, 14, and 15. 7. If the number is a square number, then it has to be the sum of successive odd numbers starting from 1. Example: For 32 = 9, the sum of successive odd numbers from 1 will be 1+3+5 = 9. Note: If a natural number cannot be expressed as a sum of successive odd natural numbers starting with 1, then it is not a perfect square. 8. Square number can be summation of two consecutive natural numbers. Example: 52 = 25 = 12 + 13; 72 = 49 = 24 + 25, etc. 9. Product of two consecutive even or odd natural numbers. Example: 11 × 13 = (12-1) × (12+1) = 122 – 1 So, in general (a+1) × (a-1) = a2 – 1. Exercise 3.2 Question 2. What will be the unit digit of the squares of the following numbers? (iii) 329 (iv) 643 (vii) 68327 (viii) 95628 (ix) 99880 Solution: Question 3. The following numbers are obviously not perfect. Give reason. (iv) 46292 (v) 74000 Solution: Question 4. The square of which of the following numbers would be an odd number or an even number? Why? (iii) 8267 (iv) 37916 Solution: Question 5. How many natural numbers lie between the square of the following numbers? (ii) 90 and 91 Solution: Question 6. Without adding, find the sum. (ii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 Solution: Question 7. (ii) 121 as the sum of 11 odd numbers. Solution: Question 8. Express the following as the sum of two consecutive integers: (iii) 472 Solution: Question 9. Find the squares of the following numbers without actual multiplication: (iii) 86 (iv) 94 Solution: Question 10. Find the squares of the following numbers containing 5 in unit’s place: (ii) 305 (iii) 525 Solution: Question 11. Write a Pythagorean triplet whose one number is (iii) 63 (iv) 80 Solution: Question 12. Observe the following pattern and find the missing digits: 212 = 441 2012 = 40401 20012 = 4004001 200012 = 4 – – – 4 – – – 1 2000012 = ————– Solution: Question 14. Observe the following pattern and find the missing digits: 72 = 49 672 = 4489 6672 = 444889 66672 = 44448889 666672 = 4 ———–8 ————– 9 6666672 = 4———–8————8 – Solution: Do the following questions Question 1. Write five numbers which you can decide by looking at their one’s digit that they are not square numbers. Question 2. What will be the unit digit of the squares of the following numbers? (i) 951 (ii) 502 (v) 5124 (vi)7625 (x) 12796 Question 3. The following numbers are obviously not perfect. Give reason. (i) 567 (ii) 2453 (iii) 5298 Question 4. The square of which of the following numbers would be an odd number or an even number? Why? (i) 573 (ii) 4096 Question 5. How many natural numbers lie between the square of the following numbers? (i) 12 and 13 Question 6. Without adding, find the sum. (i) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 Question 7. (i) Express 64 as the sum of 8 odd numbers. Question 8. Express the following as the sum of two consecutive integers: (i) 192 (ii) 332 Question 9. Find the squares of the following numbers without actual multiplication: (i) 31 (ii) 42 Question 10. Find the squares of the following numbers containing 5 in unit’s place: (i) 45 Question 11. Write a Pythagorean triplet whose one number is (i) 8 (ii) 15 Question 13. Observe the following pattern and find the missing digits: 92 = 81 992 = 9801 9992 = 998001 99992 = 99980001 999992 = 9——–8———01 9999992 = 9——–0———1 Finding the Square of a Number: 1. A number can be divided into two parts, such that the square of those numbers are known. Thus, x2 = (a +b)2, where (a + b) = x and values of square of a & b are known. Example: 252 = (20 + 5) 2 = 202 + 20 x 5 + 5 x 20 + 52 = 400 + 100 +100 + 25 = 625. 2. For numbers ending with 5, follow (a5) 2 = a x (a+1) x 100 + 25 Example: 352 = 3 x (3+1) x 100 +25 = 1225 3. Pythagorean triplets If sum of two square numbers results into a square number, then all these three numbers form a Pythagorean triplet. Example 1: 32 + 42 = 9 + 16 = 25 = 52, so 3, 4 and 5 is known as Pythagorean Triplet. In general, for any natural number m > 1, we have (2m) 2 + (m2 – 1)2 = (m2 + 1)2. So, 2m, m2 – 1 and m2 + 1 forms a Pythagorean triplet. Finding the Square Root of a Number: Square root is the inverse operation of squaring. The positive square root of a number is denoted by the symbol √. Example: √9 = 3. It cannot be -3. 1. Repeated subtraction: In this method, given square number is subtracted from successive odd natural numbers starting from 1 until result of subtraction does not become 0. Example 1: Find √16. Solution: (1) 16 – 1 =15 (2) 15 – 3 = 12 (3) 12 – 5 = 7 (4) 7 – 7 = 0 We can see the result is zero at the fourth step. Thus, √16 = 4. 2. Prime Factorization Let us understand this method by an example. Example: Find √324 using prime factorization. Solution: The prime factors of 324 = 2 x 2 x 3 x 3 x 3 x 3. Note: The prime factors of any square number exist in pair. Exercise 3.3 Question 1. By repeated subtraction of odd numbers starting from 1, find whether the following numbers are perfect squares or not? If the number is a perfect square then find its square root: (iii) 36 (iv) 90 Solution: Question 2. Find the square roots of the following numbers by prime factorisation method: (iii) 1849 (iv) 4356 (v) 6241 (viii) 9025 Solution: Question 3. Find the square roots of the following numbers by prime factorisation method: Solution: Question 4. For each of the following numbers, find the smallest natural number by which it should be multiplied so as to get a perfect square. Also, find the square root of the square number so obtained: (iii) 2178 (iv) 3042 (v) 6300 Solution: Question 5. For each of the following numbers, find the smallest natural number by which it should be divided so that this quotient is a perfect square. Also, find the square root of the square number so obtained: (iii) 3380 (iv) 16224 (v) 61347 Solution: Question 6. Find the smallest square number that is divisible by each of the following numbers: (ii) 6, 9, 27, 36 (iii) 4, 7, 8, 16 Solution: Question 7. 4225 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row. Solution: Question 9. In a school a P.T. teacher wants to arrange 2000 students in the form of rows and columns for P.T. display. If the number of rows is equal to number of columns and 64 students could not be accommodated in this arrangement. Find the number of rows. Solution: Question 11. The product of two numbers is 7260. If one number is 15 times the other number, find the numbers. Solution: Question 13. The perimeter of two squares is 60 metres and 144 metres respectively. Find the perimeter of another square equal in area to the sum of the first two squares. Solution: Do the following questions Question 1. By repeated subtraction of odd numbers starting from 1, find whether the following numbers are perfect squares or not? If the number is a perfect square then find its square root: (i) 121 (ii) 55 Question 2.
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