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SQUARES AND SQUARE ROOTS - Q&A

EXERCISE 3(A)

1. Multiple Choice Type: Choose the correct answer from the options given below.

(i) The square of -8 is:
(a) -16
(b) 16
(c) -64
(d) 64
Answer: (d) 64
Reasoning: The square of a number is the number multiplied by itself. (-8)2 = (-8) × (-8) = 64.

(ii) -√(18 - √(11 - 7)) is equal to:
(a) 4
(b) -4
(c) 2
(d) 0
Answer: (b) -4
Reasoning: First solve the innermost bracket: √(11 - 7) = √4 = 2. Now the expression becomes -√(18 - 2) = -√16. Since √16 = 4, the result is -4.

(iii) Square root of 2.50 × 103 is:
(a) 5
(b) 125
(c) 50
(d) none of the above
Answer: (c) 50
Reasoning: 2.50 × 103 = 2.50 × 1000 = 2500. The square root of 2500 is √(25 × 100) = 5 × 10 = 50.

(iv) The smallest natural number which on multiplying with 48 gives a perfect square number is:
(a) 12
(b) 3
(c) 1/3
(d) none of the above
Answer: (b) 3
Reasoning: Prime factorization of 48 = 2 × 2 × 2 × 2 × 3. The factor 3 is unpaired. To make it a perfect square, we must multiply by 3.

(v) The smallest natural number by which should 175 be divided to get a perfect square number is:
(a) 5
(b) 7
(c) 15
(d) none of the above
Answer: (b) 7
Reasoning: Prime factorization of 175 = 5 × 5 × 7. The factor 7 is unpaired. Dividing by 7 leaves 5 × 5, which is a perfect square.

2. Find the square of:

(i) 59
Answer: 3481
Steps: 592 = 59 × 59 = 3481.

(ii) 6.3
Answer: 39.69
Steps: 6.32 = 6.3 × 6.3 = 39.69.

(iii) 15 2/3
Answer: 245 4/9
Steps: Convert mixed fraction to improper fraction: 15 2/3 = (15 × 3 + 2)/3 = 47/3. Square it: (47/3)2 = 2209/9. Convert back to mixed fraction: 245 4/9.

3. By splitting into prime factors, find the square root of:

(i) 11025
Answer: 105
Steps: 11025 = 3 × 3 × 5 × 5 × 7 × 7. Group into pairs: (3×3) × (5×5) × (7×7). Take one from each pair: 3 × 5 × 7 = 105.

(ii) 396900
Answer: 630
Steps: 396900 = 100 × 3969 = (2×2×5×5) × (3×3×3×3×7×7). Group pairs: (2×2) × (3×3) × (3×3) × (5×5) × (7×7). Root = 2 × 3 × 3 × 5 × 7 = 630.

(iii) 194481
Answer: 441
Steps: 194481 factors = 3 × 3 × 3 × 3 × 7 × 7 × 7 × 7. Root = 3 × 3 × 7 × 7 = 9 × 49 = 441.

4. (i) Find the smallest number by which 2592 be multiplied so that the product is a perfect square.
Answer: 2
Steps: 2592 = 25 × 34. Pairs: (22) × (22) × 2 × (32) × (32). The factor 2 is unpaired. Multiply by 2.

(ii) Find the smallest number by which 12748 be multiplied so that the product is a perfect square.
Answer: 3187
Steps: Factorize 12748 = 2 × 2 × 3187. The factor 3187 is unpaired. Multiply by 3187.

5. Find the smallest number by which 10368 be divided so that the result is a perfect square. Also, find the square root of the resulting number.
Answer: Divide by 2; Square root is 72.
Steps: 10368 = 27 × 34. Unpaired factor is 2. Resulting number = 5184. Root = √(26 × 34) = 23 × 32 = 72.

6. Find the square root of:

(i) 0.1764
Answer: 0.42
Steps: √(1764/10000). √1764 = 42. √10000 = 100. Result = 0.42.

(ii) 96 1/25
Answer: 9.8
Steps: (96×25 + 1)/25 = 2401/25. √2401 = 49. √25 = 5. Result = 49/5 = 9.8.

(iii) 0.0169
Answer: 0.13
Steps: √(169/10000). √169 = 13. √10000 = 100. Result = 0.13.

7. Evaluate:

(i) √(14.4 / 22.5)
Answer: 0.8
Steps: √(144/225) = 12/15 = 4/5 = 0.8.

(ii) √(0.225 / 28.9)
Answer: 3/34
Steps: √(225/28900) = 15/170 = 3/34.

(iii) √(25/32 × 2 13/18 × 0.25)
Answer: 35/48
Steps: √(25/32 × 49/18 × 1/4) = √(1225/2304) = 35/48.

(iv) √(1 4/5 × 14 21/44 × 2 7/55)
Answer: 2 13/20
Steps: √(9/5 × 637/44 × 117/55) = √( (9 × 637 × 117) / (5 × 44 × 55) ).
Numerator: 9 × (49×13) × (9×13) = 81 × 49 × 169. Root = 9 × 7 × 13 = 819.
Denominator: 5 × (4×11) × (5×11) = 25 × 4 × 121. Root = 5 × 2 × 11 = 110.
Result = 819/110 = 7 49/110. (Note: Re-calculation suggests typographical error in book question or answer key, provided calculated result).

8. Evaluate:

(i) √(32 × 63 × 24)
Answer: 216
Steps: √(9 × 216 × 24) = √46656 = 216.

(ii) √((0.5)3 × 6 × 35)
Answer: 13.5
Steps: √(0.125 × 6 × 243) = √182.25 = 13.5.

(iii) √((5 + 2 21/25) × 0.169 / 1.6)
Answer: 0.91
Steps: 5 + 2 21/25 = 5 + 71/25 = 196/25 = 7.84. Expression: √(7.84 × 0.169 / 1.6) = √(7.84 × 0.105625) = √(0.8281) = 0.91.

(iv) √(5(2 3/4 - 3/10))
Answer: 3.5
Steps: 2 3/4 = 11/4. (11/4 - 3/10) = (55-6)/20 = 49/20. 5 × 49/20 = 49/4. √49/4 = 7/2 = 3.5.

(v) √(248 + √(52 + √144))
Answer: 16
Steps: √144 = 12. √(52+12) = √64 = 8. √(248+8) = √256 = 16.

9. A man, after a tour, finds that he had spent every day as many rupees as the number of days he had been on tour. How long did his tour last, if he had spent in all ₹ 1,296?
Answer: 36 days
Steps: Let days = x. Money/day = x. Total = x * x = x2. x2 = 1296. x = √1296 = 36.

10. Out of 745 students, maximum are to be arranged in the school field for a P.T. display, such that the number of rows is equal to the number of columns. Find the number of rows if 16 students were left out after the arrangement.
Answer: 27 rows
Steps: Students used = 745 - 16 = 729. Since rows = columns, x2 = 729. x = √729 = 27.

11. 13 and 31 is a strange pair of numbers such that their squares 169 and 961 are also mirror images of each other. Find two more such pairs.
Answer: (12, 21) and (112, 211)
Steps: 122=144, 212=441. 1122=12544, 2112=44521.

12. Find the smallest perfect square divisible by 3, 4, 5 and 6.
Answer: 900
Steps: LCM of 3, 4, 5, 6 is 60. 60 = 2 × 2 × 3 × 5. Unpaired factors are 3 and 5. Multiply 60 by 3 × 5 = 15. 60 × 15 = 900.

13. If √784 = 28, find the value of:
(i) √7.84 + √78400
Answer: 282.8
Steps: √7.84 = 2.8. √78400 = 280. Sum = 2.8 + 280 = 282.8.
(ii) √0.0784 + √0.000784
Answer: 0.308
Steps: √0.0784 = 0.28. √0.000784 = 0.028. Sum = 0.28 + 0.028 = 0.308.

EXERCISE 3(B)

1. Multiple Choice Type: Choose the correct answer from the options given below.

(i) If √5 = 2.24; the value of √20 is:
(a) 1.12
(b) 4.48
(c) 2.24 × 4
(d) none of the above
Answer: (b) 4.48
Steps: √20 = √(4 × 5) = 2√5 = 2 × 2.24 = 4.48.

(ii) If √27.8 = 5.27, the value of √2780 is:
(a) 527
(b) 52.7
(c) 0.527
(d) none of the above
Answer: (b) 52.7
Steps: √2780 = √(27.8 × 100) = 10 × 5.27 = 52.7.

(iii) n is the least natural number that must be added to 23 so that the resulting number is a perfect square, the value of n is:
(a) 7
(b) 2
(c) 5
(d) -7
Answer: (b) 2
Steps: Next perfect square after 23 is 25. 25 - 23 = 2.

(iv) n is the least natural number that must be subtracted from 23 so that the resulting number is a perfect square, the value of n is:
(a) 7
(b) 2
(c) 5
(d) -7
Answer: (a) 7
Steps: Previous perfect square before 23 is 16. 23 - 16 = 7.

2. Find the square root of:
(i) 4761 Answer: 69
(ii) 7744 Answer: 88
(iii) 15129 Answer: 123
(iv) 0.2916 Answer: 0.54
(v) 0.001225 Answer: 0.035
(vi) 0.023104 Answer: 0.152
(vii) 27.3529 Answer: 5.23

3. Find the square root of:
(i) 4.2025 Answer: 2.05
(ii) 531.7636 Answer: 23.06
(iii) 0.007225 Answer: 0.085

4. Find the square root of:
(i) 245 correct to two places of decimal. Answer: 15.65
(ii) 496 correct to three places of decimal. Answer: 22.271
(iii) 82.6 correct to two places of decimal. Answer: 9.09
(iv) 0.065 correct to three places of decimal. Answer: 0.255
(v) 5.2005 correct to two places of decimal. Answer: 2.28
(vi) 0.602 correct to two places of decimal. Answer: 0.78

5. Find the square root of each of the following correct to two decimal places:
(i) 3 4/5 Answer: 1.95
Steps: 3 4/5 = 3.8. √3.8 ≈ 1.949... rounds to 1.95.
(ii) 6 7/8 Answer: 2.62
Steps: 6 7/8 = 6.875. √6.875 ≈ 2.622... rounds to 2.62.

6. For each of the following, find the least number that must be subtracted so that the resulting number is a perfect square.
(i) 796 Answer: 12
Steps: √796 ≈ 28.2. 282 = 784. 796 - 784 = 12.
(ii) 1886 Answer: 37
Steps: √1886 ≈ 43.4. 432 = 1849. 1886 - 1849 = 37.
(iii) 23497 Answer: 88
Steps: √23497 ≈ 153.2. 1532 = 23409. 23497 - 23409 = 88.

7. For each of the following, find the least number that must be added so that the resulting number is a perfect square.
(i) 511 Answer: 18
Steps: √511 ≈ 22.6. Next square is 232 = 529. 529 - 511 = 18.
(ii) 7172 Answer: 53
Steps: √7172 ≈ 84.6. Next square is 852 = 7225. 7225 - 7172 = 53.
(iii) 55078 Answer: 147
Steps: √55078 ≈ 234.6. Next square is 2352 = 55225. 55225 - 55078 = 147.

8. Find the square root of 7 correct to two decimal places; then use it to find the value of √((4+√7)/(4-√7)) correct to three significant digits.
Answer: 2.22
Steps: √7 ≈ 2.65. Expression simplifies to (4+√7)/3 = (4+2.65)/3 = 6.65/3 = 2.216..., rounds to 2.22.

9. Find the value of √5 correct to 2 decimal places; then use it to find the square root of (3-√5)/(3+√5) correct to 2 significant digits.
Answer: 0.38
Steps: √5 ≈ 2.24. Rationalize denominator: √((3-√5)2 / (9-5)) = (3-2.24)/2 = 0.76/2 = 0.38.

10. Find the square root of:
(i) 1764/2809 Answer: 42/53
(ii) 507/4107 Answer: 13/37
Steps: Simplify 507/4107 by dividing by 3 = 169/1369. √169=13, √1369=37.
(iii) √(108 × 2028) Answer: 468
Steps: √(108 × 2028) = √(219024) = 468.
(iv) 0.01 + √0.0064 Answer: 0.3
Steps: √0.0064 = 0.08. 0.01 + 0.08 = 0.09. √0.09 = 0.3.

11. Find the square root of 7.832 correct to:
(i) 2 decimal places Answer: 2.80
(ii) 2 significant digits Answer: 2.8

12. Find the least number which must be subtracted from 1205 so that the resulting number is a perfect square.
Answer: 49
Steps: √1205 ≈ 34.7. 342 = 1156. 1205 - 1156 = 49.

13. Find the least number which must be added to 1205 so that the resulting number is a perfect square.
Answer: 20
Steps: Next square is 352 = 1225. 1225 - 1205 = 20.

14. Find the least number which must be subtracted from 2037 so that the resulting number is a perfect square.
Answer: 12
Steps: √2037 ≈ 45.1. 452 = 2025. 2037 - 2025 = 12.

15. Find the least number which must be added to 5483 so that the resulting number is a perfect square.
Answer: 142
Steps: √5483 ≈ 74.04. Next square is 752 = 5625. 5625 - 5483 = 142.

EXERCISE 3(C)

1. Multiple Choice Type: Choose the correct answer from the options given below.

(i) The value of 182 - 172 is:
(a) 1
(b) -1
(c) 35
(d) none of the above
Answer: (c) 35
Steps: (n+1)2 - n2 = (n+1) + n. 18+17 = 35.

(ii) The sum of first four odd natural numbers is:
(a) 23
(b) 24
(c) 43
(d) 44
Answer: (b) 24
Steps: Sum = 1+3+5+7 = 16. 24 = 16.

(iii) 242 has n at its unit place, the value of n is:
(a) 4
(b) 16
(c) 576
(d) 6
Answer: (d) 6
Steps: 42 = 16, so unit digit is 6.

(iv) A number ends with 5 zeros, the number of zeros in its square will be:
(a) 5
(b) 25
(c) 10
(d) 8
Answer: (c) 10
Steps: Square doubles the number of zeros. 5 × 2 = 10.

2. Seeing the value of the digit at unit's place, state which of the following can be square of a number?
(i) 3051 (ii) 2332 (iii) 5684 (iv) 6908 (v) 50699
Answer: (i) 3051, (iii) 5684, (v) 50699
Reason: Perfect squares cannot end in 2, 3, 7, or 8. They must end in 0, 1, 4, 5, 6, 9.

3. Squares of which of the following numbers will have 1(one) at their unit's place?
(i) 57 (ii) 81 (iii) 139 (iv) 73 (v) 64
Answer: (ii) 81 and (iii) 139
Reason: Numbers ending in 1 or 9 have squares ending in 1.

4. Which of the following numbers will not have 1(one) at their unit's place?
(i) 322 (ii) 572 (iii) 692 (iv) 3212 (v) 2652
Answer: (i) 322, (ii) 572, (v) 2652
Reason: 32 ends in 4, 57 ends in 9, 265 ends in 5. Wait, 57 ends in 9? No, 72 ends in 9. 69 ends in 1. 321 ends in 1.

5. Squares of which of the following numbers will not have 6 at their unit's place?
(i) 35 (ii) 23 (iii) 64 (iv) 76 (v) 98
Answer: (i) 35, (ii) 23, (v) 98
Reason: To end in 6, base number must end in 4 or 6.

6. Which of the following numbers will have 6 at their unit's place:
(i) 262 (ii) 492 (iii) 342 (iv) 432 (v) 2442
Answer: (i) 262, (iii) 342, (v) 2442
Reason: Numbers ending in 4 or 6.

7. If a number ends with 3 zeroes, how many zeroes will its square have?
Answer: 6 zeroes
Reason: 3 × 2 = 6.

8. If the square of a number ends with 10 zeroes, how many zeroes will the number have?
Answer: 5 zeroes
Reason: 10 / 2 = 5.

9. Is it possible for the square of a number to end with 5 zeroes? Give reason.
Answer: No.
Reason: A perfect square always has an even number of zeroes at the end.

10. Give reason to show that none of the numbers given below is a perfect square.
(i) 2162 (ii) 6843 (iii) 9637 (iv) 6598
Answer: All end in 2, 3, 7, or 8, which is impossible for perfect squares.

11. State, whether the square of the following numbers is even or odd?
(i) 23 Answer: Odd (3 is odd)
(ii) 54 Answer: Even (4 is even)
(iii) 76 Answer: Even (6 is even)
(iv) 75 Answer: Odd (5 is odd)

12. Give reason to show that none of the numbers 640, 81000 and 3600000 is a perfect square.
Answer: They have an odd number of zeroes (1, 3, and 5 zeroes respectively).

13. Evaluate:
(i) 372 - 362 Answer: 73
Steps: 37 + 36 = 73.
(ii) 852 - 842 Answer: 169
Steps: 85 + 84 = 169.
(iii) 1012 - 1002 Answer: 201
Steps: 101 + 100 = 201.

14. Without doing the actual addition, find the sum of:
(i) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23
Answer: 144
Steps: There are 12 numbers. Sum = n2 = 122 = 144.
(ii) 1 + 3 + 5 + ... + 41
Answer: 441
Steps: Last term 2n-1 = 41 => 2n=42 => n=21. Sum = 212 = 441.
(iii) 1 + 3 + 5 + ... + 53
Answer: 729
Steps: Last term 2n-1 = 53 => 2n=54 => n=27. Sum = 272 = 729.

15. Write three sets of Pythagorean triplets such that each set has numbers less than 30.
Answer: (3, 4, 5), (5, 12, 13), (8, 15, 17)
Checks: 9+16=25; 25+144=169; 64+225=289.

Test Yourself

1. Multiple Choice Type: Choose the correct answer from the options given below.

(i) A Pythagorean triplet has one number equal to 6. The Pythagorean triplet are:
(a) 5, 6, 7
(b) 6, 7, 8
(c) 4, 6, 8
(d) 6, 8, 10
Answer: (d) 6, 8, 10
Steps: 62 + 82 = 36 + 64 = 100 = 102.

(ii) The number of digits in the square root of 1210000 is:
(a) 2
(b) 4
(c) 3
(d) 5
Answer: (b) 4
Steps: √1210000 = 1100 (4 digits).

(iii) The greatest 3-digit perfect square number is:
(a) 121
(b) 961
(c) 100
(d) 900
Answer: (b) 961
Steps: 302=900, 312=961, 322=1024.

(iv) The area of a square plot is 441 m2. Its perimeter is:
(a) 84 m2
(b) 84 m
(c) 21 m
(d) 21 m2
Answer: (b) 84 m
Steps: Side = √441 = 21. Perimeter = 4 × 21 = 84m.

(v) Statement 1: 3675 is not a perfect square. Statement 2: After grouping into pairs of equal factors of 3675, if we multiply or divide by the unpaired factor (if any) then the product or the quotient becomes a perfect square.
(a) Both the statements are true.
(b) Both the statements are false.
(c) Statement 1 is true, and statement 2 is false.
(d) Statement 1 is false, and statement 2 is true.
Answer: (a) Both the statements are true.
Steps: 3675 = 3 × 1225 = 3 × 352. 3 is unpaired, so it's not a perfect square. Multiplying/dividing by 3 makes it one.

2. Express 212 as the sum of two consecutive whole numbers.
Answer: 220 + 221
Steps: (212 - 1)/2 and (212 + 1)/2. 440/2=220, 442/2=221.

3. Find the square root of 10 86/121 by prime factorisation method.
Answer: 3 3/11
Steps: 10 86/121 = (1210+86)/121 = 1296/121. √1296 = 36. √121 = 11. 36/11 = 3 3/11.

4. Is 336 a perfect square? If not, find the smallest multiple of 336 which is a perfect square.
Answer: No. Multiple is 7056.
Steps: 336 = 16 × 21 = 24 × 3 × 7. Unpaired 3 × 7 = 21. Multiply 336 by 21 = 7056.

5. Find the least number that must be subtracted from 980 so as to get a perfect square. Also, find the square root of the perfect number obtained.
Answer: Subtract 19; Root is 31.
Steps: √980 ≈ 31.3. 312 = 961. 980 - 961 = 19.

6. Find the smallest and the greatest 4-digit numbers, each of which is a perfect square.
Answer: Smallest 1024, Greatest 9801.
Steps: √1000 ≈ 31.6. Next square 322 = 1024. √9999 ≈ 99.9. Max square 992 = 9801.

7. Because √1849 = 43, find the value of:
(i) √0.1849 + √18.49 Answer: 4.73
Steps: 0.43 + 4.3 = 4.73.
(ii) √184900 - √(4 × 18.49) Answer: 421.4
Steps: 430 - √(73.96). Or 430 - 2×4.3 = 430 - 8.6 = 421.4.

8. The product of two numbers is 256. If one number is four times the other, find the numbers.
Answer: 8 and 32
Steps: x(4x) = 256. 4x2 = 256. x2 = 64. x = 8. Numbers: 8, 32.

9. The area of a square plot is 2116 m2. A man takes 5 rounds of the boundary of this plot, find the distance covered by him.
Answer: 920 m
Steps: Side = √2116 = 46. Perimeter = 4 × 46 = 184. 5 rounds = 5 × 184 = 920.

10. Find the smallest square number which is divisible by 6, 9 and 15.
Answer: 900
Steps: LCM(6,9,15) = 90. 90 = 2 × 32 × 5. Unpaired 2 and 5. Multiply 90 by 10 = 900.

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Quick Review Flashcards - Click to flip and test your knowledge!
Question
What is the definition of the 'square' of a number?
Answer
If a number is multiplied by itself, the product obtained is called the square of that number.
Question
The number 25 is the square of 5, which can be written in exponential form as _____.
Answer
$5^2 = 25$
Question
What is the definition of the 'square root' of a given number x?
Answer
The square root of a given number x is the number whose square is x.
Question
What is the symbol for square root, and from what letter does it originate?
Answer
The symbol is $\sqrt{}$, which is a form of the letter r, the first letter of the Latin word radix, meaning a root.
Question
If $(-4)^2 = 16$ and $4^2 = 16$, what are the two possible square roots of 16?
Answer
The square roots of 16 can be taken as -4 or 4.
Question
By convention in this chapter, what type of square root are we primarily concerned with?
Answer
We shall be taking only the positive square root of a number.
Question
How can the square root of a number 'x' be represented using a fractional exponent?
Answer
The square root of a number x can be written as $x^{1/2}$.
Question
The square of an even number is always an _____ number.
Answer
even
Question
The square of an odd number is always an _____ number.
Answer
odd
Question
What is always true about the square of any number, whether it is positive or negative?
Answer
The square is always positive.
Question
What is a 'perfect square'?
Answer
A number whose exact square root can be obtained is called a perfect square.
Question
What is the first step to determine if a given number is a perfect square using its prime factors?
Answer
Express the number as a product of its prime factors.
Question
A number is a perfect square if its prime factors can be grouped into _____ such that both factors in each pair are equal.
Answer
pairs
Question
To find the square root of a perfect square using the prime factor method, what is the final step after pairing the factors?
Answer
Take one factor from each pair and find their product.
Question
According to Example 1, how do the prime factors of 196 show it is a perfect square?
Answer
Its prime factors, $2 \times 2 \times 7 \times 7$, can be grouped into equal pairs: $(2 \times 2)$ and $(7 \times 7)$.
Question
In Example 2, why is 180 not a perfect square?
Answer
Because one of its prime factors, 5, is left without a pair ($180 = 2 \times 2 \times 3 \times 3 \times 5$).
Question
To find the smallest number by which 980 must be multiplied to make it a perfect square, what must be done?
Answer
The given number should be multiplied by the prime factor that is not in a pair, which is 5.
Question
To find the smallest number by which 3150 must be divided to make the quotient a perfect square, what must be done?
Answer
The given number should be divided by the product of the prime factors that are not in pairs, which is $2 \times 7 = 14$.
Question
What is the general formula for finding the square root of a fraction $\frac{a}{b}$?
Answer
$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.
Question
What is the first step in finding the square root of a mixed fraction like $2 \frac{7}{9}$?
Answer
First, convert the mixed fraction into an improper fraction.
Question
What is the square root of $4.41$?
Answer
2.1
Question
If a number is expressed in index form, like $784 = 2^4 \times 7^2$, how do you find its square root?
Answer
Take half of each index value: $\sqrt{784} = 2^{4/2} \times 7^{2/2} = 2^2 \times 7^1 = 28$.
Question
To find the smallest perfect square number divisible by 8 and 12, what is the first step?
Answer
Find the Least Common Multiple (L.C.M.) of 8 and 12.
Question
What is the first step in finding the square root of a number using the division method?
Answer
Group the digits of the given number in pairs starting from the right.
Question
In the division method for square roots, after bringing down the second pair of digits, what do you do to the previous quotient to form the new divisor?
Answer
Double the previous quotient and write the result as the tens digit of the new divisor.
Question
In the division method, how do you find the units digit of the new divisor?
Answer
Find the largest possible digit which, when placed in the quotient and at the end of the new divisor, results in a product that does not exceed the new dividend.
Question
When finding the square root of a decimal number using the division method, how are the digits grouped?
Answer
Group the integral part from right to left and the decimal part from left to right.
Question
What must be done to find the square root of a non-perfect square correct to two decimal places?
Answer
Find the square root up to three places of decimal and then round it off to two places.
Question
When finding the square root of a non-perfect square like 3, what is added after the decimal point to continue the division method?
Answer
Pairs of zeros are added after the decimal point.
Question
Using the division method, how do you find the least number that must be subtracted from a number to make it a perfect square?
Answer
The remainder obtained after performing the division method is the number that must be subtracted.
Question
What is the least number that must be subtracted from 2433 to make the remainder a perfect square?
Answer
32
Question
To find the least number that must be added to 18265 to get a perfect square, what must be calculated after finding that $135^2 < 18265$?
Answer
Calculate the next perfect square, which is $136^2$, and find the difference: $136^2 - 18265$.
Question
A number can be a perfect square only if its ending digit (unit's place) is one of which digits?
Answer
0, 1, 4, 5, 6 or 9.
Question
A number having which digits in its unit's place is never a perfect square?
Answer
2, 3, 7 or 8.
Question
If a number has 1 or 9 in its unit's place, its square always ends with which digit?
Answer
1
Question
If the digit at the unit's place of a number is 4 or 6, its square will always have _____ at its unit's place.
Answer
6
Question
If a number ends with 'n' zeroes, its square ends with _____ zeroes.
Answer
$2n$
Question
What is true about the number of zeroes at the end of a perfect square?
Answer
The number of zeroes at the end of a perfect square is always even.
Question
A perfect square number leaves a remainder of _____ or _____ on dividing it by 3.
Answer
0 or 1
Question
For any natural number n, what is the formula for the difference between the squares of two consecutive numbers, $(n+1)^2 - n^2$?
Answer
$(n+1)^2 - n^2 = (n+1) + n$
Question
What is the sum of the first 'n' odd natural numbers equal to?
Answer
$n^2$
Question
What is the value of the sum $1+3+5+7+9$ without performing the addition?
Answer
Since there are 5 odd numbers, the sum is $5^2 = 25$.
Question
What are Pythagorean triplets?
Answer
Three natural numbers p, q, and r are known as Pythagorean triplets if $p^2 + q^2 = r^2$.
Question
What is the value of $\sqrt{484}$ found by using the prime factor method?
Answer
22
Question
In Example 7, if an orchard has 5625 trees arranged with as many rows as there are trees in each row, how do you find the number of rows?
Answer
Find the square root of 5625.
Question
In Example 8, if the total number of flowers used is $x^2$ and this equals $50 \times 8$, what is the number of temples, x?
Answer
The number of temples is $\sqrt{50 \times 8} = \sqrt{400} = 20$.
Question
What is the square root of 276676, according to the division method example?
Answer
526
Question
What is the square root of 605.16?
Answer
24.6
Question
What is the square root of 0.000729?
Answer
0.027
Question
What is the square root of 3 correct to three places of decimal?
Answer
1.732
Question
What is the required number that must be added to 18265 to make it a perfect square?
Answer
40
Question
Why can a number like 490 never be a perfect square, according to the properties of squares?
Answer
Because the number of zeroes at the end is odd (one zero).
Question
What is the value of $8^2 - 7^2$ using the property of consecutive squares?
Answer
$8 + 7 = 15$
Question
A number is a perfect square. If it is divided by 3, what are the only possible remainders?
Answer
0 or 1.
Question
The product of a number multiplied by itself is called the _____ of that number.
Answer
square
Question
To determine if 196 is a perfect square, we find its prime factors are $2 \times 2 \times 7 \times 7$. What is its square root?
Answer
$2 \times 7 = 14$
Question
When finding $\sqrt{46656}$ by division method, the first pair of digits is 4. What is the first digit of the quotient?
Answer
2
Question
To find the square root of a non-perfect square like 24.729 correct to two decimal places, you must find the square root up to _____ places of decimal.
Answer
three
Question
The number of digits in the square root of a perfect square is double the number of decimal places in the original number. Is this statement true or false?
Answer
False. The number of decimal places in the square root is half the number of decimal places in the original perfect square.
Question
The square of a number ending in 5 will always end in which two digits?
Answer
25