Root of a number doubt

Question

how to find root of 58.97 ?

Please tell me any shortcut method (if any) ?

Answer 1

1. There is an algorithm that can be used to find square roots
2. Group the numbers from right to left in pairs
3. In this case, the two groups would be 58 and 97
4. Starting from the leftmost group (58), find the largest number whose square is less than or equal to 58.  That number is 7.  And since 7*7 = 49, write down the number 49 in the row directly under the number in the square root.  Also, write down the 7 above the square root sign, since we know this is the first integer in the solution.
5. Create a left hand column, and write down the number 7 on the first row of this left hand column.
6. Now, to find the next number, double the 7.  This gives you 14.  Write that down on the second row of the left hand column.
7. Subtract the 49 from 58.  This give you 9.  Bring down the next pair of numbers, i.e, 97.  So you now have 9.97.   Think of this number as 997 (i.e.., ignore the decimal point)
8. Think of the number 14 on the left hand side as 14_.   Notice the underscore.  You now have to figure out the largest number to replace the underscore, which - when multiplied by that number - will give you a quotient that is less than 9.97.  That number is 6, because when you multiply 146 * 6, you get 876.   And 876 < 997.
9. So 6 is the next number in our solution.  Write that above the square root sign, next to the 7.  Bring the decimal point up.  So you now have 7.6 as the start of your answer.
10. To get more accuracy, add two trailing zeroes to the number under the square root sign.  So you now have 58.97.00. 
11. When you subtract 8.76 from 9.97, you get 1.21.  Bring down those trailing zeroes, and you get 1.2100
12. Now double your current answer, which sits at 7.6.  If you double 7.6, you get 15.2.  Again, ignore the fact that there is a decimal point here, and write down 152 on the next row of the left hand column.
13. Now make that 152_.  Find the largest number to substitute for the underscore, which - when multiplied by itself - will give you a quotient less than 12100.  That number is 7.  This is because 1527 * 7 = 10689, and 10689 < 12100.  If you had used 8, then 1528 * 8 = 12224, which is larger than 12100.  So we know the right number to use is 7.
14. Now add the 7 as the next number above the square root sign, and this gives you 7.67
15. You can continue on this way by adding more and more pairs of trailing zeroes, but i will stop here.  And so your answer is 7.67, accurate to 2 decimal places.
16. Obviously, this method has lost some of its charm since the advent of calculators.

          

                             7.67

                   ______________

7       |       √58.97.00.00

14     |          49.

146   |      --------------------

         |            9.97

         |            8.76

1527  |    -------------------

                      1.2100

                      1.0689

               --------------------

                        1411.00

Answer 2

1.  Olden days: get log of the number, divide it by two, and look up what number has THAT logarithm..required using interpolation and printed table of logarithms.

 

2.  Current technology: press square root key on calculator.

 

3.  By hand: trial & error can work, just keep estimating, squaring, and making refined correction, and repeat ad nauseum.

 

#2 is BEST.