How do you find the square root of 96?

Answer 1

Exactly is impossible as it is an irrational number.

However, I can find an approximation by:

• Using my calculator

Depending upon which calculator I either press:

• [√][9][6][=]; or
• [9][6][√]

• Using my slide rule

Line the cursor up on the D (or C) scale on 96 and read the [approximation of the] square root off the A (or B) scale.

• Using logs

Using logs to any base, find the logarithm of 96 (either using tables, slide rule or calculator), divide it by 2 and then find the anti-logarithm of that (again using tables, slide rule or calculator)

• By a kind of "long" division

1. Write the number in pairs of digits from the right hand end starting with the units;
2. add a decimal point and write pairs of zeros after the decimal point to the required degree of accuracy;
3. Find the largest square that is not greater than the first pair of digits;
4. write the square root of this above the pair and write the square below the pair and subtract (just like normal long division);
5. Bring down the next pair of digits;
6. Double the result so far (ignoring the decimal point), multiply it by 10 and write it as the divisor next to the result of step 4;
7. Replace the last zero digit with a digit so that when the divisor is multiplied by this digit, it is the largest number not greater than the result of step 5;
8. write this under the result of step 5 and subtract;
9. repeat from step 5 until all the digits have been used up and the result of the subtraction is zero (and there are only pairs of zeros after the decimal point in the original number remaining), or the required degree of accuracy has been found.

Doing the "long" division for 96, calculating to 4 dp to get result rounded to 3 dp:

................ 9 . . 7 . 9 . 7 . 9

.............-------------------

.............| 96 . 00 00 00 00

.............. 81 ....................... largest square ≤ 96 is 92 = 81; put 9 in result (over 96)

.............. --- ...................... write 81 underneath and subtract;

..... 187 | 15 . 00 ................ bring down next pair of digits; double 9 and x10 to get 180

.............. 13 . 09 ................ 187 x 7 = 1309, 188 x 8 = 1504, so make units 7, put in result (over 00)

.............. ------- ................. write 1309 underneath and subtract;

... 1949 | . 1 . 91 00 ........... bring down next pair of digits; double 97 and x 10 to get 1940

................ 1 . 75 41 ........... 1949 x 9 = 17541; making units 9, put in result (over 00);

................ --------- ............ write 17541 underneath and subtract;

.19587 | ...... 15 59 00 ...... bring down next pair of digits; double 979 and x 10 to get 19580

......................13 71 09 ...... 19587 x 7 = 137109; making units 7, put in result (over 00);

......................--------- ....... write 137109 underneath and subtract;

195949 | ........ 1 87 91 00 . bring down next pair of digits; double 9797 and x10 to get 195940

........................ 1 76 35 41 . 195949 x 9 = 1763541; making the units 9, put in result (over 00)

........................ ----------- . write 1763541 underneath and subtract;

........................... 11 55 59 . stop as to get ans to 3 dp work to 4 dp and round using 4th dp.

The final subtraction is not really necessary as only the 4th decimal place digit is required to round to 3 dp

To 3 dp, √96 ≈ 9.798