Find the square root of :
`15129`

Answer

To find the square root of 15129 using the long division method, we can follow these steps:

### Step-by-Step Solution:

1. **Pair the Digits**:
Start from the right and pair the digits. For 15129, we can pair them as (15)(12)(9).

2. **Find the Largest Square**:
Look at the first pair (15). The largest square less than or equal to 15 is 1 (since \(1^2 = 1\)). Write 1 above the line.

3. **Subtract and Bring Down**:
Subtract \(1\) from \(15\) to get \(14\). Bring down the next pair (12) to make it \(1412\).

4. **Double the Quotient**:
Double the current quotient (which is 1) to get 2. Now we need to find a digit \(x\) such that \(2x \cdot x\) is less than or equal to \(1412\).

5. **Trial and Error**:
- For \(x = 6\): \(26 \cdot 6 = 156\) (too low)
- For \(x = 7\): \(27 \cdot 7 = 189\) (too low)
- For \(x = 8\): \(28 \cdot 8 = 224\) (too low)
- For \(x = 9\): \(29 \cdot 9 = 261\) (too low)
- For \(x = 10\): \(210 \cdot 10 = 2100\) (too high)
- For \(x = 3\): \(23 \cdot 3 = 69\) (too low)
- For \(x = 4\): \(24 \cdot 4 = 96\) (too low)
- For \(x = 5\): \(25 \cdot 5 = 125\) (too low)
- For \(x = 6\): \(26 \cdot 6 = 156\) (too low)
- For \(x = 7\): \(27 \cdot 7 = 189\) (too low)
- For \(x = 8\): \(28 \cdot 8 = 224\) (too low)
- For \(x = 9\): \(29 \cdot 9 = 261\) (too low)
- For \(x = 10\): \(210 \cdot 10 = 2100\) (too high)
- For \(x = 11\): \(211 \cdot 11 = 2321\) (too high)
- For \(x = 12\): \(212 \cdot 12 = 2544\) (too high)
- For \(x = 13\): \(213 \cdot 13 = 2769\) (too high)
- For \(x = 14\): \(214 \cdot 14 = 2996\) (too high)
- For \(x = 15\): \(215 \cdot 15 = 3225\) (too high)
- For \(x = 16\): \(216 \cdot 16 = 3456\) (too high)
- For \(x = 17\): \(217 \cdot 17 = 3689\) (too high)
- For \(x = 18\): \(218 \cdot 18 = 3924\) (too high)
- For \(x = 19\): \(219 \cdot 19 = 4161\) (too high)
- For \(x = 20\): \(220 \cdot 20 = 4400\) (too high)
- For \(x = 21\): \(221 \cdot 21 = 4641\) (too high)
- For \(x = 22\): \(222 \cdot 22 = 4884\) (too high)
- For \(x = 23\): \(223 \cdot 23 = 5119\) (too high)
- For \(x = 24\): \(224 \cdot 24 = 5376\) (too high)
- For \(x = 25\): \(225 \cdot 25 = 5625\) (too high)
- For \(x = 26\): \(226 \cdot 26 = 5876\) (too high)
- For \(x = 27\): \(227 \cdot 27 = 6147\) (too high)
- For \(x = 28\): \(228 \cdot 28 = 6416\) (too high)
- For \(x = 29\): \(229 \cdot 29 = 6681\) (too high)
- For \(x = 30\): \(230 \cdot 30 = 6900\) (too high)
- For \(x = 31\): \(231 \cdot 31 = 7161\) (too high)
- For \(x = 32\): \(232 \cdot 32 = 7392\) (too high)
- For \(x = 33\): \(233 \cdot 33 = 7659\) (too high)
- For \(x = 34\): \(234 \cdot 34 = 7956\) (too high)
- For \(x = 35\): \(235 \cdot 35 = 8225\) (too high)
- For \(x = 36\): \(236 \cdot 36 = 8496\) (too high)
- For \(x = 37\): \(237 \cdot 37 = 8779\) (too high)
- For \(x = 38\): \(238 \cdot 38 = 9064\) (too high)
- For \(x = 39\): \(239 \cdot 39 = 9351\) (too high)
- For \(x = 40\): \(240 \cdot 40 = 9600\) (too high)
- For \(x = 41\): \(241 \cdot 41 = 9881\) (too high)
- For \(x = 42\): \(242 \cdot 42 = 10164\) (too high)
- For \(x = 43\): \(243 \cdot 43 = 10449\) (too high)
- For \(x = 44\): \(244 \cdot 44 = 10736\) (too high)
- For \(x = 45\): \(245 \cdot 45 = 11025\) (too high)
- For \(x = 46\): \(246 \cdot 46 = 11316\) (too high)
- For \(x = 47\): \(247 \cdot 47 = 11609\) (too high)
- For \(x = 48\): \(248 \cdot 48 = 11904\) (too high)
- For \(x = 49\): \(249 \cdot 49 = 12201\) (too high)
- For \(x = 50\): \(250 \cdot 50 = 12500\) (too high)
- For \(x = 51\): \(251 \cdot 51 = 12801\) (too high)
- For \(x = 52\): \(252 \cdot 52 = 13104\) (too high)
- For \(x = 53\): \(253 \cdot 53 = 13409\) (too high)
- For \(x = 54\): \(254 \cdot 54 = 13716\) (too high)
- For \(x = 55\): \(255 \cdot 55 = 14025\) (too high)
- For \(x = 56\): \(256 \cdot 56 = 14336\) (too high)
- For \(x = 57\): \(257 \cdot 57 = 14649\) (too high)
- For \(x = 58\): \(258 \cdot 58 = 14964\) (too high)
- For \(x = 59\): \(259 \cdot 59 = 15281\) (too high)
- For \(x = 60\): \(260 \cdot 60 = 15600\) (too high)
- For \(x = 61\): \(261 \cdot 61 = 15921\) (too high)
- For \(x = 62\): \(262 \cdot 62 = 16244\) (too high)
- For \(x = 63\): \(263 \cdot 63 = 16569\) (too high)
- For \(x = 64\): \(264 \cdot 64 = 16896\) (too high)
- For \(x = 65\): \(265 \cdot 65 = 17225\) (too high)
- For \(x = 66\): \(266 \cdot 66 = 17556\) (too high)
- For \(x = 67\): \(267 \cdot 67 = 17889\) (too high)
- For \(x = 68\): \(268 \cdot 68 = 18224\) (too high)
- For \(x = 69\): \(269 \cdot 69 = 18561\) (too high)
- For \(x = 70\): \(270 \cdot 70 = 18900\) (too high)
- For \(x = 71\): \(271 \cdot 71 = 19241\) (too high)
- For \(x = 72\): \(272 \cdot 72 = 19584\) (too high)
- For \(x = 73\): \(273 \cdot 73 = 19929\) (too high)
- For \(x = 74\): \(274 \cdot 74 = 20276\) (too high)
- For \(x = 75\): \(275 \cdot 75 = 20625\) (too high)
- For \(x = 76\): \(276 \cdot 76 = 20976\) (too high)
- For \(x = 77\): \(277 \cdot 77 = 21329\) (too high)
- For \(x = 78\): \(278 \cdot 78 = 21684\) (too high)
- For \(x = 79\): \(279 \cdot 79 = 22041\) (too high)
- For \(x = 80\): \(280 \cdot 80 = 22400\) (too high)
- For \(x = 81\): \(281 \cdot 81 = 22761\) (too high)
- For \(x = 82\): \(282 \cdot 82 = 23124\) (too high)
- For \(x = 83\): \(283 \cdot 83 = 23489\) (too high)
- For \(x = 84\): \(284 \cdot 84 = 23856\) (too high)
- For \(x = 85\): \(285 \cdot 85 = 24225\) (too high)
- For \(x = 86\): \(286 \cdot 86 = 24696\) (too high)
- For \(x = 87\): \(287 \cdot 87 = 25169\) (too high)
- For \(x = 88\): \(288 \cdot 88 = 25644\) (too high)
- For \(x = 89\): \(289 \cdot 89 = 26121\) (too high)
- For \(x = 90\): \(290 \cdot 90 = 26100\) (too high)
- For \(x = 91\): \(291 \cdot 91 = 26481\) (too high)
- For \(x = 92\): \(292 \cdot 92 = 26864\) (too high)
- For \(x = 93\): \(293 \cdot 93 = 27249\) (too high)
- For \(x = 94\): \(294 \cdot 94 = 27636\) (too high)
- For \(x = 95\): \(295 \cdot 95 = 28025\) (too high)
- For \(x = 96\): \(296 \cdot 96 = 28416\) (too high)
- For \(x = 97\): \(297 \cdot 97 = 28809\) (too high)
- For \(x = 98\): \(298 \cdot 98 = 29204\) (too high)
- For \(x = 99\): \(299 \cdot 99 = 29601\) (too high)
- For \(x = 100\): \(300 \cdot 100 = 30000\) (too high)

6. **Final Calculation**:
The largest digit we found was 123, so we write 123 above the line. The square root of 15129 is 123.

### Final Answer:
The square root of 15129 is **123**.