Squares and Square Roots, Cube and Cube Roots

Q1: Find the square root of:

i. 2.89

Step 1: Pair digits: (2)(89)
Step 2: Apply division method:

     1.7
    ------
 1 | 2.89
   - 1        ← 1 × 1 = 1
   ------
     1 89     ← bring down 89
     
 27 | 189     ← 2×1=2
     -189     → 27×7=189 
   ------
        0
  

Answer: √2.89 = 1.7

ii. 33.64

Step 1: Pair digits: (33)(64)
Step 2: Apply division method:

     5.8
    ------
5 |  33.64
   - 25        ← 5×5 = 25
   ------
      8 64     ← bring down 64
      
 108 | 864     ← 2×5 = 10 
      -864	   → 108×8 = 864 
   --------
         0
  

Answer: √33.64 = 5.8

iii. 156.25

Step 1: Pair digits: (156)(25)
Step 2: Apply division method:

     12.5
    ------
1 | 156.25
   -1         ← 1×1 = 1
   -------
     56			← bring down
     
 22 | 56		← 1×2= 2
     -44		→ 22×2 = 44
   -------
      1225      ← bring down
      
  245 | 1225     ← 2×12 = 24
       -1225     → 245×5 = 1225
   --------
         0
  

Answer: √156.25 = 12.5

iv. 75.69

Step 1: Pair digits: (75)(69)
Step 2: Apply division method:

     8.7
    ------
 8| 75.69
  - 64         ← 8×8 = 64
   ------
    11 69      ← bring down
    
 167 | 1169    ← 2×8 = 16
      -1169    → 167×7 = 1169 
   ---------
         0
  

Answer: √75.69 = 8.7

v. 5.4289

Step 1: Pair digits: (5)(42)(89)
Step 2: Apply division method:

     2.33
    -------
 2| 5.4289
  - 4         ← 2×2 = 4
   ------
    1 42      ← bring down
    
 43 | 142      ← 2×2 = 4
     -129	   → 43×3 = 129
   ------
       13 89   ← bring down
       
   463|1389    ← 2×23 = 46 
      -1389    → 463×3 = 1389
   -------
         0
  

Answer: √5.4289 = 2.33

vi. 18.1476

Step 1: Pair digits: (18)(14)(76)
Step 2: Apply division method:

     4.26
    -------
 4 | 18.1476
   - 16        ← 4×4 = 16
   ------
      2 14     ← bring down
      
   82|214      ← 2×4 = 8 
      164      → 82×2 = 164
   ------
         50 76   ← bring down
         
   846 | 5076      ← 2×42 = 84
        -5076      → 846×6 = 5076
       -------
         0
  

Answer: √18.1476 = 4.26

Q2: Evaluate \(\sqrt2\) up to two places of decimal.

Step 1: Pair the digits of 2 as: (2.00 00 00)
Since we want the answer up to 2 decimal places, we take 2 decimal pairs.
Step 2: Apply division method step-by-step:

      1.414
     ------
 1 | 2.000000
   - 1        ← 1 × 1 = 1
   ------
     1 00     ← bring down 00
     
  24 | 100    ← 2×1 = 2 
       -96    ⇒ 24×4 = 96 
     ------
         4 00    ← bring down next pair
       
  281 | 400   ← 2×14 = 28
       -281    ⇒ 281×1 = 281
    ------
        119 00     ← bring down 00
     
  2824 | 11900    ← 2×141 = 282 
        -11296     ⇒ 2824×4 = 11296
       ------
          604    ← bring down next pair
  

Step 3: Approximate value up to two decimal places = 1.414
Note: Since next digit is 4, we do not round up here, we take accurate long division cutoff.
Answer: \(\sqrt{2} = 1.41\)

Q3: Evaluate \(\sqrt{0.4}\) correct up to two places of decimal.

Step 1: Place decimal properly and convert 0.4 into suitable form:
We write: \(0.4 = 0.4000\) (add pairs of zeroes for precision)
Now pair the digits from the decimal point: (0.40)(00)(00)
Step 2: Apply square root division method:

       0.632
     --------
0 | 0.400000
  - 0             ← 0 × 0 = 0
   ------
      40          ← bring down first pair after decimal
      
 06 | 40       	← Try 0×2 = 0
     -36		⇒ 6×6 = 36
   ------
      4 00     ← bring down next pair
      
 123 | 400       ← 2×6 = 12
      -369		⇒ Try 123×3 = 369
     ------
        31 00	← bring down next pair
      
 1262 | 3100       ← 2×63 = 126
       -2524		⇒ 1262×2 = 2524
       ------
         576 		→ Stop here (only need up to 2 decimal places)
  

Step 3: Approximate value up to two decimal places = 0.632
Note: Since next digit is 2, we do not round up here, we take accurate long division cutoff.
Answer:\(\sqrt{0.4} \approx 0.63\)

Q4: Evaluate \(\sqrt{2.8}\) correct up to two places of decimal.

Step 1: Convert 2.8 into long division format:
We write: \(2.8 = 2.80 00\), grouping digits in pairs from the decimal.Step 2: Apply long division method:

       1.673
     --------
1 | 2.8000
   -1        ← 1×1 = 1
   ------
    1.80     ← bring down next pair
    
 26 | 180      ← 2×1 = 2
     -156   		⇒ Try 26×6 = 156
     ------
       24 00    ← bring down next pair
      
  327 | 2400    ← 2×16 = 32
       -2289	⇒ 327×7 = 12289
       ------
         111 00 ← bring down next pair
      
   3343 | 11100    ← 2×167 = 334
         -10029		⇒ 3343×3 = 100029
         ------
           1071 	→ ignore extra digits
  

Step 3: Approximate value up to two decimal places = 1.673
Note: Since next digit is 3, we do not round up here, we take accurate long division cutoff.
Answer:\(\sqrt{2.8} \approx 1.67\)

Q5: Find the length of each side of a square whose area is equal to the area of a rectangle of length 13.6 metres and breadth 3.4 metres.

Step 1: Find the area of the rectangle \[ \text{Area} = \text{Length} \times \text{Breadth} = 13.6 \times 3.4 \]Step 2: Multiply the numbers \[ 13.6 \times 3.4 = 46.24 \, \text{sq. metres} \]Step 3: Let the side of the square be \( x \) \[ x^2 = 46.24 \Rightarrow x = \sqrt{46.24} \]Now, find \( \sqrt{46.24} \) using the division method.
Step 4: Make digit pairs: (46)(24)(00)…

       6.8
     --------
6 | 46.24
   -36        ← 6×6 = 36
   ------
    10.24     ← bring down next pair
    
 128 | 1024      ← 2×6 = 12
     - 1024   		⇒ 128×8 = 1024
     ------
       0  

Step 5: The square root of 46.24 is 6.8
Answer: The length of each side of the square is 6.8 metres.

Q6; If \(\sqrt{1+\frac{27}{169}}=1+\frac{x}{13}\), find the value of x.

Step 1: Add the fractions inside the square root \[ \sqrt{1 + \frac{27}{169}} = \sqrt{\frac{169}{169} + \frac{27}{169}} = \sqrt{\frac{196}{169}} \]Step 2: Take square roots of numerator and denominator \[ \sqrt{\frac{196}{169}} = \frac{\sqrt{196}}{\sqrt{169}} = \frac{14}{13} \]So the equation becomes: \[ \frac{14}{13} = 1 + \frac{x}{13} \]Step 3: Subtract 1 from both sides \[ \frac{14}{13} – \frac{13}{13} = \frac{x}{13} \Rightarrow \frac{1}{13} = \frac{x}{13} \]Step 4: Equating the numerators \[ x = 1 \]Answer: \(x = 1\)

Q7: Find the square root of:

i. \(\sqrt{3\frac{13}{36}}\)

Step 1: Convert to improper fraction: \[ 3\frac{13}{36} = \frac{(3 \times 36) + 13}{36} = \frac{108 + 13}{36} = \frac{121}{36} \]Step 2: Take square root: \[ \sqrt{\frac{121}{36}} = \frac{\sqrt{121}}{\sqrt{36}} = \frac{11}{6} \]Answer: \(\sqrt{3\frac{13}{36}} = \frac{11}{6}\)

ii. \(\sqrt{4\frac{73}{324}}\)

Step 1: Convert to improper fraction: \[ 4\frac{73}{324} = \frac{(4 \times 324) + 73}{324} = \frac{1296 + 73}{324} = \frac{1369}{324} \]Step 2: Take square root: \[ \sqrt{\frac{1369}{324}} = \frac{\sqrt{1369}}{\sqrt{324}} = \frac{37}{18} \]Answer: \(\sqrt{4\frac{73}{324}} = \frac{37}{18}\)

iii. \(\sqrt{3\frac{942}{2209}}\)

Step 1: Convert to improper fraction: \[ 3\frac{942}{2209} = \frac{(3 \times 2209) + 942}{2209} = \frac{6627 + 942}{2209} = \frac{7569}{2209} \]Step 2: Take square root: \[ \sqrt{\frac{7569}{2209}} = \frac{\sqrt{7569}}{\sqrt{2209}} = \frac{87}{47} \]Answer: \(\sqrt{3\frac{942}{2209}} = \frac{87}{47}\)

iv. \(\sqrt{\frac{1089}{4624}}\)

Step 1: Take square root directly: \[ \sqrt{\frac{1089}{4624}} = \frac{\sqrt{1089}}{\sqrt{4624}} = \frac{33}{68} \]Answer: \(\sqrt{\frac{1089}{4624}} = \frac{33}{68}\)

Q8: Find the value of:

i. \(\frac{\sqrt{243}}{\sqrt{867}}\)

Step 1: Use the identity \[ \frac{\sqrt{243}}{\sqrt{867}} = \sqrt{\frac{243}{867}} \]Step 2: Simplify the fraction \[ \frac{243}{867} = \frac{81 \times 3}{289 \times 3} = \frac{81}{289} \]Step 3: Take square root \[ \sqrt{\frac{81}{289}} = \frac{\sqrt{81}}{\sqrt{289}} = \frac{9}{17} \]Answer: \(\frac{\sqrt{243}}{\sqrt{867}} = \frac{9}{17}\)

ii. \(\frac{\sqrt{1183}}{\sqrt{2023}}\)

Step 1: Use the identity \[ \frac{\sqrt{1183}}{\sqrt{2023}} = \sqrt{\frac{1183}{2023}} \]Step 2: Simplify the fraction \[ \frac{1183}{2023} = \frac{81 \times 7}{289 \times 7} = \frac{169}{289} \]Step 3: Take square root \[ \sqrt{\frac{169}{289}} = \frac{\sqrt{169}}{\sqrt{289}} = \frac{13}{17} \] Answer: \(\frac{\sqrt{1183}}{\sqrt{2023}} = \frac{13}{17}\)

Q9: Find the value of \(\sqrt{15625}\) and hence evaluate \(\sqrt{156.25}+\sqrt{1.5625}\).

Step 1: Use the division method to find \(\sqrt{15625}\)

       125
     --------
1 | 1 56 25
   -1        ← 1×1 = 1
   ------
    0 56     ← bring down next pair
    
 22 | 56      ← 2×1 = 2
     -44   		⇒ Try 22×2 = 44
     ------
       12 25    ← bring down next pair
      
  245 | 1225    ← 2×12 = 24
       -1225	⇒ 245×5 = 1225
       ------
           0

Step 2: Move decimal →
\(\sqrt{156.25} = \frac{\sqrt{15625}}{10} = \frac{125}{10} = 12.5\)
\(\sqrt{1.5625} = \frac{\sqrt{15625}}{100} = \frac{125}{100} = 1.25\)
Step 3: Add both values: \[ 12.5 + 1.25 = 13.75 \]Answer: \(\sqrt{156.25} + \sqrt{1.5625} = 13.75\)

Q10: Evaluate:

i. \(\sqrt{99} \times \sqrt{396}\)

Step 1: Use the property of square roots \[ \sqrt{99} \times \sqrt{396} = \sqrt{99 \times 396} \]Step 2: Multiply the numbers inside the square root \[ 99 \times 396 = 39204 \]Step 3: Now take the square root of 39204 \[ \sqrt{39204} = 198 \]Answer: \(\sqrt{99} \times \sqrt{396} = 198\)

ii. \(\sqrt{147} \times \sqrt{243}\)

Step 1: Use the property of square roots \[ \sqrt{147} \times \sqrt{243} = \sqrt{147 \times 243} \]Step 2: Multiply the numbers inside the square root \[ 147 \times 243 = 35721 \]Step 3: Now take the square root of 35721 \[ \sqrt{35721} = 189 \]Answer: \(\sqrt{147} \times \sqrt{243} = 189\)

Q11: Evaluate:

i. \(\sqrt{\frac{0.289}{0.00121}}\)

Step 1: Convert decimals to fractions \[ 0.289 = \frac{289}{1000}, \quad 0.00121 = \frac{121}{100000} \]Step 2: Divide the fractions \[ \frac{0.289}{0.00121} = \frac{\frac{289}{1000}}{\frac{121}{100000}} = \frac{289 \times 100000}{1000 \times 121} = \frac{28900000}{121000} \]Step 3: Simplify the fraction \[ \frac{28900000}{121000} = \frac{28900}{121} \]Step 4: Take the square root \[ \sqrt{\frac{28900}{121}} = \frac{\sqrt{28900}}{\sqrt{121}} = \frac{170}{11} \]Answer: \(\sqrt{\frac{0.289}{0.00121}} = \frac{170}{11}\)

ii. \(\sqrt{\frac{48.4}{0.289}}\)

Step 1: Convert decimals to fractions \[ 48.4 = \frac{484}{10}, \quad 0.289 = \frac{289}{1000} \]Step 2: Divide the fractions \[ \frac{48.4}{0.289} = \frac{\frac{484}{10}}{\frac{289}{1000}} = \frac{484 \times 1000}{10 \times 289} = \frac{484000}{2890} \]Step 3: Simplify the fraction \[ \frac{484000}{2890} = \frac{48400}{289} \] Step 4: Take the square root \[ \sqrt{\frac{48400}{289}} = \frac{\sqrt{48400}}{\sqrt{289}} = \frac{220}{17} \]Answer: \(\sqrt{\frac{48.4}{0.289}} = \frac{220}{17}\)

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

Step 1: Find \(\sqrt{0.0064} = 0.08\)Step 2: Add 0.01 and 0.08 \[ 0.01 + 0.08 = 0.09 \]Step 3: Now, find the square root of 0.09 \[ \sqrt{0.09} = 0.3 \]Answer: \(\sqrt{0.01 + \sqrt{0.0064}} = 0.3\)

iv. \(\sqrt{0.01} + \sqrt{0.81} + \sqrt{1.21} + \sqrt{0.0009}\)

Step 1: Find each square root: \[ \sqrt{0.01} = 0.1, \quad \sqrt{0.81} = 0.9, \quad \sqrt{1.21} = 1.1, \quad \sqrt{0.0009} = 0.03 \]Step 2: Add all the values \[ 0.1 + 0.9 + 1.1 + 0.03 = 2.13 \]Answer: \(\sqrt{0.01} + \sqrt{0.81} + \sqrt{1.21} + \sqrt{0.0009} = 2.13\)

v. \(\sqrt{41-\sqrt{21+\sqrt{19-\sqrt9}}}\)

Step 1: Find \(\sqrt{9} = 3\) \[ 19 – 3 = 16 \quad \Rightarrow \quad \sqrt{16} = 4 \]Step 2: Now, find \(\sqrt{21 + 4} = \sqrt{25} = 5\) \[ 41 – 5 = 36 \quad \Rightarrow \quad \sqrt{36} = 6 \]Answer: \(\sqrt{41 – \sqrt{21 + \sqrt{19 – \sqrt{9}}}} = 6\)

vi. \(\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}}\)

Step 1: Find \(\sqrt{225} = 15\) \[ \sqrt{154 + 15} = \sqrt{169} = 13 \]Step 2: Find \(\sqrt{108 + 13} = \sqrt{121} = 11\) \[ \sqrt{25 + 11} = \sqrt{36} = 6 \]Step 3: Find \(\sqrt{10 + 6} = \sqrt{16} = 4\)Answer: \(\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}} = 4\)

Q12: Three-fifths of the square of a certain number is 126.15. Find the number.

Step 1: We are given the equation: \[ \frac{3}{5} \times x^2 = 126.15 \]Step 2: Multiply both sides by 5 to eliminate the fraction: \[ 3 \times x^2 = 126.15 \times 5 \]\[ 3 \times x^2 = 630.75 \]Step 3: Divide both sides by 3: \[ x^2 = \frac{630.75}{3} = 210.25 \]Step 4: Now take the square root of both sides: \[ x = \sqrt{210.25} = 14.5 \]Answer: The number is \( 14.5 \)