Question

Find the square root of 14641 by repeated subtraction.

Answer 1

Hint:
Here, we will find the square root of the given number in the repeated subtraction method by subtracting the successive odd integers from the number. This process of subtraction continues until the difference becomes zero. The step at which the result is obtained is known as the square root of the number.

Complete step by step solution:
We are given the number 14641 . We will find the square root of the number by repeated subtraction.
We know that the property of square numbers states that if a natural number is a square number, then it has to written as the sum of successive odd numbers starting from the first odd integer as 1.
We will start the repeated subtraction method by subtracting the odd integer 1 from the number. We will continue the process with consecutive odd integers.
\[\begin{array}{l}14641 - 1 = 14640\\14640 - 3 = 14637\\14637 - 5 = 14632\\14632 - 7 = 14625\\14625 - 9 = 14616\end{array}\]
 \[\begin{array}{l}14616 - 11 = 14605\\14605 - 13 = 14592\\14592 - 15 = 14577\\14577 - 17 = 14560\\14560 - 19 = 14541\end{array}\]
 \[\begin{array}{l}14541 - 21 = 14520\\14520 - 23 = 14497\\14497 - 25 = 14472\\14472 - 27 = 14445\\14445 - 29 = 14416\end{array}\]
 \[\begin{array}{l}14416 - 31 = 14385\\14385 - 33 = 14352\\14352 - 35 = 14317\\14317 - 37 = 14280\\14280 - 39 = 14241\end{array}\]
 \[\begin{array}{l}14241 - 41 = 14200\\14200 - 43 = 14157\\14157 - 45 = 14112\\14112 - 47 = 14065\\14065 - 49 = 14016\end{array}\]
 \[\begin{array}{l}14016 - 51 = 13965\\13965 - 53 = 13912\\13912 - 55 = 13857\\13857 - 57 = 13800\\13800 - 59 = 13741\end{array}\]
 \[\begin{array}{l}13741 - 61 = 13680\\13680 - 63 = 13617\\13617 - 65 = 13552\\13552 - 67 = 13485\\13485 - 69 = 13416\end{array}\]
 \[\begin{array}{l}13416 - 71 = 13345\\13345 - 73 = 13272\\13272 - 75 = 13197\\13197 - 77 = 13120\\13120 - 79 = 13041\end{array}\]
 \[\begin{array}{l}13041 - 81 = 12960\\12960 - 83 = 12877\\12877 - 85 = 12792\\12792 - 87 = 12705\\12705 - 89 = 12616\end{array}\]
 \[\begin{array}{l}12616 - 91 = 12525\\12525 - 93 = 12432\\12432 - 95 = 12337\\12337 - 97 = 12240\\12240 - 99 = 12141\end{array}\]
 \[\begin{array}{l}12141 - 101 = 12040\\12040 - 103 = 11937\\11937 - 105 = 11832\\11832 - 107 = 11725\\11725 - 109 = 11616\end{array}\]
 \[\begin{array}{l}11616 - 111 = 11505\\11505 - 113 = 11392\\11392 - 115 = 11277\\11277 - 117 = 11160\\11160 - 119 = 11041\end{array}\]
 \[\begin{array}{l}11041 - 121 = 10920\\10920 - 123 = 10797\\10797 - 125 = 10672\\10672 - 127 = 10545\\10545 - 129 = 10416\end{array}\]
 \[\begin{array}{l}10416 - 131 = 10285\\10285 - 133 = 10152\\10152 - 135 = 10017\\10017 - 137 = 9880\\9880 - 139 = 9741\end{array}\]
 \[\begin{array}{l}9741 - 141 = 9600\\9600 - 143 = 9457\\9457 - 145 = 9312\\9312 - 147 = 9165\\9165 - 149 = 9016\end{array}\]
 \[\begin{array}{l}9016 - 151 = 8865\\8865 - 153 = 8712\\8712 - 155 = 8557\\8557 - 157 = 8400\\8400 - 159 = 8241\end{array}\]
 \[\begin{array}{l}8241 - 161 = 8080\\8080 - 163 = 7917\\7917 - 165 = 7752\\7752 - 167 = 7585\\7585 - 169 = 7416\end{array}\]
 \[\begin{array}{l}7416 - 171 = 7245\\7245 - 173 = 7072\\7072 - 175 = 6897\\6897 - 177 = 6720\\6720 - 179 = 6541\end{array}\]
 \[\begin{array}{l}6541 - 181 = 6360\\6360 - 183 = 6177\\6177 - 185 = 5992\\5992 - 187 = 5805\\5805 - 189 = 5616\end{array}\]
 \[\begin{array}{l}5616 - 191 = 5425\\5425 - 193 = 5232\\5232 - 195 = 5037\\5037 - 197 = 4840\\4840 - 199 = 4641\end{array}\]
 \[\begin{array}{l}4641 - 201 = 4440\\4440 - 203 = 4237\\4237 - 205 = 4032\\4032 - 207 = 3825\\3825 - 209 = 3616\end{array}\]
 \[\begin{array}{l}3616 - 211 = 3405\\3405 - 213 = 3192\\3192 - 215 = 2977\\2977 - 217 = 2760\\2760 - 219 = 2541\end{array}\]
\[\begin{array}{l}2541 - 221 = 2320\\2320 - 223 = 2097\\2097 - 225 = 1872\\1872 - 227 = 1645\\1645 - 229 = 1416\end{array}\]
\[\begin{array}{l}1416 - 231 = 1185\\1185 - 233 = 952\\952 - 235 = 717\\717 - 237 = 480\\480 - 239 = 241\\241 - 241 = 0\end{array}\]
Thus, we got the result as \[0\] in the \[121th\] step, so the square root of the number \[14641\] is \[121\] i.e., \[\sqrt {14641} = 121\]

Therefore, the square root of the number \[14641\] is \[121\].

Note:
We can also find the square root of the number by prime factorization method, long division method, number line method and average method. Prime factorization method and Repeated Subtraction method is applicable only when the given number is a perfect square number. We should always remember that the square root of a number is the step at which zero is obtained and not the number which is subtracted.