Squares and Square Roots

Q1: Multiple Choice Type

i. If \(\sqrt5=2.24\); the value of \(\sqrt{20}\) is

Step 1:
We know: \[ 20 = 4 \times 5 \Rightarrow \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2 \times 2.24 = 4.48 \]Answer: b. 4.48

ii. If \(\sqrt{27.8}=5.27\), the value of \(\sqrt{2780}\) is:

Step 1: \[ 2780 = 27.8 \times 100 \Rightarrow \sqrt{2780} = \sqrt{27.8 \times 100} = \sqrt{27.8} \times \sqrt{100} = 5.27 \times 10 = 52.7 \]Answer: b. 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:

Step 1:
Let’s test perfect squares greater than 23:
25 − 23 = 2 (25 is a perfect square, 5² = 25)
Answer: b. 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:

Step 1:
Let’s test perfect squares less than 23:
16 is the nearest square below 23 → 23 − 16 = 7
Answer: a. 7

Q2: Find square root of:

i. 4761

Step-by-step using division method:

    69
  _______
6| 47 61
  -36       ← 6 × 6 = 36
  -----
   11 61      ← Bring down next pair
 
129| 1161      ← (2×6) = 12
    -1161 	  → 129×9 = 1161
    -----
      0

Answer: 69

ii. 7744

    88
  _______
8| 77 44
  -64       ← 8×8 = 64
  -----
   13 44      ← Bring down next pair
 
168| 1344      ← (2×8) = 16
    -1344      → 168×8 = 1344
     -----
       0

Answer: 88

iii. 15129

    123
  ________
1| 1 51 29
  -1
  ------
   0 51		← Bring down next pair
 
 22| 51      ← (2×1) = 2
    -44      → 2×2 = 44
  ------
      7 29      ← Bring down next pair
 
243| 729      ← (2×12) = 24
    -729      → 243×3 = 729
    ------
       0

Answer: 123

iv. 0.2916

Note: Group digits after decimal in pairs: 29|16

    0.54
  _________
5| 0. 29 16
  -0  25        ← 0×0 = 25
  ------
   0. 04 16      ← Bring down next pair
 
  104| 416      ← (2×0) = 0
      -416      → 104×4 = 416
    --------
        0

Answer: 0.54

v. 0.001225

Group digits: 00|12|25

    0.035
  ___________
0| 00 | 12 | 25
  -00
  -------
    0 12      ← Bring down next pair
 
 03| 12        ← (2×0) = 0
     -9        → 3×3 = 9 
  --------
      3 25      ← Bring down next pair
 
  65| 425      ← (2×3) = 6
     -325      → 65×5 = 325
    ------
      0 

Answer: 0.035

vi. 0.023104

Group digits: 00|23|10|40

    0.152
  _____________
1| 02 31 04
   -1
  --------
    1 31      ← Bring down next pair
 
 25| 131      ← (2×1) = 2
    -125      → 25×5 = 125
    -----
       6 04      ← Bring down next pair
 
  302| 604      ← (2×15) = 30
      -604      → 302×2 = 604
      ------
        0

Answer: 0.152

vii. 27.3529

Group: 27|35|29

    5.23
  ____________
5| 27. 35 29
  -25       ← 5×5 = 25
  ------
    2 35      ← Bring down next pair
 
102| 235      ← (2×5) = 10
    -204      → 102×2 = 204
    --------
      31 29      ← Bring down next pair
 
 1043| 3129      ← (2×52) = 104
      -3129      → 1043×3 = 3129
      -------
        0

Answer: 5.23

Q3: Find the square root:

i. 4.2025

Step-by-step: Group digits as: 04 | 20 | 25

     2.05
  ___________
2| 04 20 25
  -04         ← 2×2 = 4
  -----
    0 20     ← Bring down next pair
 
 40| 20      ← (2×2) = 4
    -00      → 40×0 = 0
    --------
     20 25     ← Bring down next pair
 
 405| 2025      ← (2×20) = 40
     -2025      → 405×5 = 2025
    --------
        0

Answer: 2.05

ii. 531.7636

Step-by-step: Group digits as: 05 | 31 | 76 | 36

     23.06
  _______________
2| 05 31. 76 36
  - 4           ← 2×2 = 4
  ------
     1 31     ← Bring down next pair
 
 43| 131      ← (2×2) = 4
    -129      → 43×3 = 129
    --------
       2 76     ← Bring down next pair
 
   460| 276      ← (2×23) = 46
       -000      → 460×0 = 000
      --------
        276 36     ← Bring down next pair
 
    4606| 27636      ← (2×230) = 460
         -27636      → 4606×6 = 4606
        --------
            0

Answer: 23.06

iii. 0.007225

Step-by-step: Group digits as: 00 | 07 | 22 | 50

     0.085
  _______________
0| 00. 00 72 25
  -00
  ------
    0 00     ← Bring down next pair
 
 00| 000      ← (2×0) = 0
    -000      → 00×0 = 0
    --------
       0 72     ← Bring down next pair
 
    08| 72      ← (2×0) = 0
       -64      → 8×8 = 64
      --------
         8 25     ← Bring down next pair
 
    165| 825      ← (2×8) = 16
        -825      → 165×5 = 825
      --------
         0

Answer: 0.085

Q4: Find the square root of:

i. 245 correct to two places of decimal.

     15.652
  _______________
1| 2 45. 00 00 00
  -1               ← 1×1 = 1
  -------
   1 45     ← Bring down next pair
 
25| 145      ← (2×1) = 2
   -125      → 25×5 = 125
   ------
     20 00     ← Bring down next pair
 
 306| 2000      ← (2×15) = 30
     -1836      → 306×6 = 1836
    --------
       164 00     ← Bring down next pair
 
  3125| 16400      ← (2×156) = 312
       -15625      → 3125×5 = 15625
       --------
          775 00     ← Bring down next pair
 
   31302| 77500      ← (2×1565) = 3130
         -62604      → 31302×2 = 62604
         --------
          14896

Answer: 15.65

ii. 496 correct to three places of decimal.

     22.2710
  __________________
2| 4 96 00 00 00
  -4              ← 2×2 = 4
  ------
   0 96     ← Bring down next pair
 
 42| 96      ← (2×2) = 4
    -84      → 42×2 = 84
    --------
     12 00     ← Bring down next pair
 
 442| 1200      ← (2×22) = 44
      -884      → 442×2 = 884
    --------
       316 00     ← Bring down next pair
 
 4447| 31600      ← (2×222) = 444
       31129      → 4447×7 = 31129
    --------
         471 00     ← Bring down next pair
 
   44541| 47100      ← (2×2227) = 4454
         -44541      → 44541×1 = 44541
         --------
            2559 00     ← Bring down next pair
 
    445420| 255900      ← (2×22271) = 44542
                -0      → 445420×0 = 0
            --------
            255900

Answer: 22.271

iii. 82.6 correct to two places of decimal.

     9.088
  _______________
9| 82. 60 00 00
  -81         ← 9×9 = 81
  ------
    1 60     ← Bring down next pair
 
 180| 160      ← (2×9) = 18
     -000      → 180×0 = 0
    --------
      160 00     ← Bring down next pair
 
 1808| 16000      ← (2×90) = 180
      -14464      → 1808×8 = 14464
    --------
        1536 00     ← Bring down next pair
 
  18168| 153600      ← (2×908) = 1816
        -145344      → 18168×8 = 145344
       --------
           8256

Answer: 9.09

iv. 0.065 correct to three places of decimal.

     0.2549
  __________________
0| 0. 06 50 00 00
  -0
  ------
    0 06     ← Bring down next pair
 
 02| 6      ← (2×0) = 0
    -4      → 2×2 = 4
    -----
     2 50     ← Bring down next pair
 
 45| 250      ← (2×2) = 4
    -225      → 45×5 = 225
    --------
      25 00     ← Bring down next pair
 
  504| 2500      ← (2×25) = 50
      -2016      → 504×4 = 2016
       --------
        484 00     ← Bring down next pair
 
   5089| 48400      ← (2×254) = 508
        -45801      → 5089×9 = 45081
       --------
          2599

Answer: 0.255

v. 5.2005 correct to two places of decimal.

     2.280
  _______________
2| 5. 20 05 00 
  -4          ← 2×2 = 4
  ------
   1 20     ← Bring down next pair
 
 42| 120      ← (2×2) = 4
     -84      → 42× = 84
    --------
      36 05     ← Bring down next pair
 
  448| 3605      ← (2×22) = 44
      -3584      → 448×8 = 3584
       --------
         21 00     ← Bring down next pair
 
   4560| 2100      ← (2×228) = 456
         -000      → 4560×0 = 0
       --------
         2100

Answer: 2.28

vi. 0.602 correct to two places of decimal.

     0.775
  _______________
0| 00. 60 20 00
  - 0
  ------
    0 60     ← Bring down next pair
 
 07| 60      ← (2×0) = 0
    -49      → 7×7 = 49
    --------
     11 20     ← Bring down next pair
 
 147| 1120      ← (2×7) = 14
     -1029      → 147×7 = 1029
    --------
        91 00     ← Bring down next pair
 
 1545| 9100      ← (2×77) = 154
      -7725      → 1545×5 = 7725
    --------
       1375

Answer: 0.78

Q5: Find the square root of each of the following correct to two decimal places:

i. \(3\frac{4}{5}\)

Step 1: Convert the mixed number into improper fraction
3 + 4/5 = (3×5 + 4)/5 = 19/5 = 3.8
Step 2: Now find √3.8 using the division method:

    1.949
  ____________
1| 3. 80 00 00
  -1          ← 1×1 = 1
  ------
   2 80       ← Bring down next pair
 
 29| 280      ← (2×1) = 2
    -261      → 29×9 = 261
    --------
      19 00       ← Bring down next pair
 
  384| 1900      ← (2×19) = 38
      -1536      → 384×4 = 1536
      --------
        364 00       ← Bring down next pair
 
  3889| 36400      ← (2×194) = 388
       -35001      → 3889×9 = 35001
        --------
         1399

Answer: 1.95

ii. \(6\frac{7}{8}\)

Step 1: Convert the mixed number into improper fraction
6 + 7/8 = (6×8 + 7)/8 = 55/8 = 6.875
Step 2: Now find √6.875 using the division method:

     2.621
  _______________
2| 6. 87 50 00
  -4           ← 2×2 = 4
  ------
   2 87       ← Bring down next pair
 
 46| 287      ← (2×2) = 4
    -276      → 46×6 = 276
    --------
      11 00       ← Bring down next pair
 
  522| 1100      ← (2×26) = 52
      -1044      → 522×2 = 1044
     --------
         56 00       ← Bring down next pair
 
 5241| 5600      ← (2×262) = 524
      -5241      → 5241×1 = 5241
    --------
        359

Answer: 2.62

Q6: For each of the following, find the least number that must be subtracted so that the resulting number is a perfect square.

i. 796

Step 1: Use the division method to find √796

   28
  _________
2| 7 96
  -4            ← 2×2 = 4
  ------
   3 96       ← Bring down next pair
 
 48| 396      ← (2×2) = 4
    -384      → 48×8 = 384
    --------
      12

Step 2: The remainder is 12
Step 3: Subtract remainder from 796
796 − 12 = 784
Answer: 12 must be subtracted. Perfect square = 784

ii. 1886

Step 1: Use the division method to find √1886

      43
   __________
4| 18 86
  -16           ← 4×4 = 16
  ------
    2 86       ← Bring down next pair
 
 83| 286      ← (4×2) = 8
    -249      → 83×3 = 249
    --------
      37 

Step 2: The remainder is 37
Step 3: Subtract remainder from 1886
1886 − 37 = 1849
Answer: 37 must be subtracted. Perfect square = 1849

iii. 23497

Step 1: Use the division method to find √23497

    153
  _____________
1| 2 34 97
  -1            ← 1×1 = 1
  ------
   1 34       ← Bring down next pair
 
 25| 134      ← (1×2) = 2
    -125      → 25×5 = 125
    --------
      997       ← Bring down next pair
 
 303| 997      ← (15×2) = 30
     -909      → 303×3 = 909
    --------
       88

Step 2: The remainder is 88
Step 3: Subtract remainder from 23497
23497 − 88 = 23409
Answer: 88 must be subtracted. Perfect square = 23436

Q7: For each of the following, find the least number that must be added so that the resulting number is a perfect square.

Use long division method to get the next square number greater than the given number. Then subtract the original number from that square.

i. 511

Step 1: Find square root of 511 using long division method

   22
  _______
2| 5 11
  -4          ← 2×2 = 4
  ------
   1 11       ← Bring down next pair
 
 42| 111      ← (2×2) = 2
     -84      → 42×2 = 84
    --------
      27

Step 2: Quotient is 22, remainder is 27
⇒ Next perfect square = (22+1)² = 23² = 529
⇒ Add = 529 − 511 = 18
Answer: 18 must be added. Perfect square = 529

ii. 7172

Step 1: Find square root of 7172 using long division method

    84
  _________
8| 71 72
  -64         ← 8×8 = 64
  ------
    7 72       ← Bring down next pair
 
164| 772      ← (8×2) = 16
    -656      → 164×4 = 656
    --------
     116

Step 2: Quotient is 84, remainder is 116
⇒ Next perfect square = (84+1)² = 85² = 7225
⇒ Add = 7225 − 7172 = 53
Answer: 53 must be added. Perfect square = 7225

iii. 55078

Step 1: Find square root of 55078 using long division method

    234
  _____________
2| 5 50 78
  -4         ← 2×2 = 4
  ------
   1 50       ← Bring down next pair
 
 43| 150      ← (2×2) = 4
    -129      → 43×3 = 129
    --------
      21 78       ← Bring down next pair
 
  464| 2178      ← (23×2) = 46
      -1856      → 464×4 = 1856
    --------
        322

Step 2: Quotient is 234, remainder is 322
⇒ Next perfect square = (234+1)² = 235² = 55225
⇒ Add = 55225 − 55078 = 147
Answer: 147 must be added. Perfect square = 55225

Q8: Find the square root of 7 correct to two decimal places; then use it to find the value of \(\sqrt{\frac{4+\sqrt7}{4-\sqrt7}}\) correct to three significant digits.

i. Find √7 correct to two decimal places

Step 1: Estimate √7 using the division method

    2.645
  ________
2| 7. 00 00
  -4               ← 2×2 = 4
  -------
   3 00       ← Bring down next pair
 
 46| 300      ← (2×2) = 4
    -276      → 46×6 = 276
    --------
      24 00       ← Bring down next pair
 
 524| 2400      ← (26×2) = 52
     -2096      → 524×4 = 2096
    --------
       304 00       ← Bring down next pair
 
 5285| 30400      ← (264×2) = 528
      -26425      → 5285×5 = 26425
      --------
        3975

So, √7 ≈ 2.645 (correct to two decimal places)
Answer: √7 ≈ 2.65

ii. Use √7 to evaluate: √[(4 + √7)/(4 − √7)] correct to 3 significant digits

Step 1: Use the identity:\[ \sqrt{\frac{(4+\sqrt7)\times(4+\sqrt7)}{(4-\sqrt7)\times(4+\sqrt7)}\ =\ \sqrt{\frac{{(4+\sqrt7)}^2}{16-7}}=\frac{4+\sqrt7}{3}}\ \]Substitute √7 ≈ 2.65:\[ \frac{4 + 2.65}{3} = \frac{6.65}{3} = 2.216\overline{6} \]Rounded to 3 significant digits:
Answer: 2.22

Q9: Find the value of \(\sqrt5\) correct to 2 decimal places; then use it to find the square root of \(\frac{3-\sqrt5}{3+\sqrt5}\) correct to 2 significant digits.

i. Find √5 correct to two decimal places

Step 1: Use division method to estimate √5

    2.236
  ________
2| 5. 00 00
  -4           → 2×2 = 4
  -------
   1 00       ← Bring down next pair
 
 42| 100      ← (2×2) = 4
     -84      → 42×2 = 84
    --------
      16 00       ← Bring down next pair
 
 443| 1600      ← (22×2) = 44
     -1329      → 443×3 = 1329
      --------
       271 00       ← Bring down next pair
 
 4466| 27100      ← (223×2) = 446
      -26796      → 4466×6 = 26796
      --------
         304

So, √5 ≈ 2.236 (correct to two decimal places)
Answer: √5 ≈ 2.24

ii. Evaluate: √[(3 − √5)/(3 + √5)] correct to 2 significant digits

Step 2: Use rationalization:\[ \sqrt{\frac{3 – \sqrt{5}}{3 + \sqrt{5}}} = \sqrt{\frac{(3 – \sqrt{5})^2}{(3 + \sqrt{5})(3 – \sqrt{5})}} = \sqrt{\frac{(3 – \sqrt{5})^2}{9 – 5}} = \sqrt{\frac{(3 – \sqrt{5})^2}{4}} = \frac{3 – \sqrt{5}}{2} \]Now substitute √5 ≈ 2.24:\[ \frac{3 – 2.24}{2} = \frac{0.76}{2} = 0.38 \]Rounded to 2 significant digits:
Answer: 0.38

Q10: Find the square root of:

i. \(\sqrt{\frac{1764}{2809}}\)

Step 1: Find √1764 using division method:

    42
  ________
4| 17 64
  -16           → 4×4 = 16
  -------
    1 64       ← Bring down next pair
 
 82| 164      ← (4×2) = 8
    -164      → 82×2 = 164
  -------
     0

⇒ √1764 = 42
Step 2: Find √2809 using division method:

    53
  ________
5| 28 09
  -25		→ 5×5 = 25
  -----
    3 09       ← Bring down next pair
 
103| 309      ← (5×2) = 10
    -309      → 103×3 = 309
  -------
      0

⇒ √2809 = 53\[ \sqrt{\frac{1764}{2809}} = \frac{√1764}{√2809} = \frac{42}{53} \]Answer: \(\frac{42}{53}\)

ii. \(\sqrt{\frac{507}{4107}}\)

Step 1: Simplify the fraction: \[ \frac{507}{4107} = \frac{507 ÷ 3}{4107 ÷ 3} = \frac{169}{1369} \]Step 2: Now find square roots:
√169 = 13

    37
  ________
3| 13 69
   -9		→ 3×3 = 9
  -----
    4 69       ← Bring down next pair
 
 67| 469      ← (3×2) = 6
    -469      → 67×7 = 469
  -------
      0

⇒ √1369 = 37\[ \sqrt{\frac{507}{4107}} = \sqrt{\frac{169}{1369}} = \frac{13}{37} \]Answer: \(\frac{13}{37}\)

iii. \(\sqrt{108 \times 2028}\)

Step 1: Multiply the numbers: \[ 108 × 2028 = 219024 \]Step 2: Find √219024 using division method:

      468
   ________
4| 21 90 24
  -16			→ 4×4 = 16
   -----
    5 90       ← Bring down next pair
 
86| 590      ← (4×2) = 8
   -516      → 86×6 = 516
    -------
     74 24       ← Bring down next pair
 
 928| 7424      ← (46×2) = 92
     -7424      → 928×8 = 7424
     -------
        0

⇒ √219024 = 468Answer: 468

iv. \(0.01 + \sqrt{0.0064}\)

Step 1: √0.0064 using division method:\[ \sqrt{0.0064} = 0.08 \]Step 2: Add to 0.01\[ 0.01 + 0.08 = 0.09 \]Answer: 0.09

Q11: Find the square root of 7.832 correct to:

i. 2 decimal places

Step 1: Use the division method to find √7.832
Group the digits in pairs from the decimal point:

   2.798
  _________
2| 7. 83 20
  -4         ← 2×2 = 4
  ------
   3 83       ← Bring down next pair
 
47| 383      ← (2×2) = 4
   -329      → 47×7 = 329
   -------
     54 20       ← Bring down next pair
 
549| 5420      ← (27×2) = 54
    -4941      → 549×9 = 4941
    -------
      479 00       ← Bring down next pair
 
5588| 47900      ← (279×2) = 558
     -44704      → 5588×8 = 44704
    -------
       3196

So, √7.832 ≈ 2.798 ≈ 2.80 (rounded to 2 decimal places)
Answer: 2.80

ii. 2 significant digits

From above, we found:
√7.832 ≈ 2.798…
Rounding 2.798 to 2 significant digits:

→ First two significant digits are “2.8”
Answer: 2.8

Q12: Find the least number which must be subtracted from 1205 so that the resulting number is a perfect square.

Step 1: Use the division method to find the square root of 1205.
We find the square root of 1205 using long division method:

      34
   _________
3| 12 05
   -9       ← 3×3 = 9
   ------
    3 05       ← Bring down next pair
 
64| 305      ← (3×2) = 6
   -256      → 64×4 = 256
   -------
     49

The quotient is 34 and remainder is 49.
So, √1205 is not a perfect square.
Step 2: Subtract 49 from 1205: \[ 1205 – 49 = 1156 = 34^2 \]Answer: 49 must be subtracted from 1205 to make it a perfect square.

Q13: Find the least number which must be added to 1205 so that the resulting number is a perfect square.

Step 1: Use the division method to find the square root of 1205.
We apply the long division method:

      34
   _________
3| 12 05
   -9       ← 3×3 = 9
   ------
    3 05       ← Bring down next pair
 
64| 305      ← (3×2) = 6
   -256      → 64×4 = 256
   -------
     49

The quotient is 34 and remainder is 49.
This tells us:
34² = 1156 and 35² = 1225
Step 2: Find the nearest perfect square greater than 1205: \[ 35^2 = 1225 \]Step 3: Subtract the original number from the next perfect square: \[ 1225 – 1205 = 20 \]Answer: 20 must be added to 1205 to make it a perfect square.

Q14: Find the least number which must be subtracted from 2037 so that the resulting number is a perfect square.

Step 1: Use long division method to find the square root of 2037.

      45
   _________
4| 20 37
 - 16         ← 4×4 = 16
 ------
    4 37       ← Bring down next pair
 
85| 437      ← (4×2) = 8
   -425      → 85×5 = 425
   -------
     12

Quotient = 45, remainder = 12
So, √2037 is not a perfect square.
Step 2: Subtract the remainder from the original number: \[ 2037 – 12 = 2025 = 45^2 \]Answer: 12 must be subtracted from 2037 to make it a perfect square.

Q15: Find the least number which must be added to 5483 so that the resulting number is a perfect square.

Step 1: Use long division method to find the square root of 5483.

    74
  _________
7| 54 83
 - 49           ← 7 × 7 = 49
 ------
    5 83       ← Bring down next pair
 
144| 583      ← (4×2) = 8
    -576      → 144×4 = 576
   -------
       7

Quotient = 74, remainder = 5
So, √5483 is not a perfect square.
Nearest perfect square greater than 5483 is: \[ 75^2 = 5625 \]Step 2: Subtract the original number from the next perfect square: \[ 5625 – 5483 = 142 \]Answer: 142 must be added to 5483 to make it a perfect square.