De-nesting nested radicals using an Excel spreadsheet

This wasn't very difficult to set up.

Let's say you are given "√(17/5 - √24/5)" i.e. you have to find the square root of "17/5 - √24/5":

The Steps are:

1. Write this down in the proper format - the "1" in the middle is implied if there is no number there. The calculations are the same whether the sign in the middle is "-" or "+", you only need this again at Step 7.

2. Write a "2" in the middle and adjust the right hand number accordingly. I set up a look-up table, so if the number in the middle in Step 1 is "1" you do 24/5 ÷ 4 = 6/5. NB, dividing by "1/4" is the same as multiplying by 4. You can reconstruct the table by doing "4 ÷ middle number squared"

3. Then we apply the same trick as for solving quadratic equations. Divide 17/5 by 2 and square it (or square 17/5 and divide the result by 4) = 289/100. Compare this with 6/5. Subtract the smaller number from the larger number = 169/100.

4. Take the square root of the result from 3. √169/100 = 13/10.

5. Divide the left hand number 17/5 (from Step 1) by 2 = 17/10 and add your answer from Step 4. 17/10 + 13/10 = 30/10 = 3.

6. Divide the left hand number 17/5 by 2 and subtract your answer from Step 4. 17/10 - 13/10 = 4/10 = 2/5.

7. Now check in the first line whether the sign in the middle was "+" or "-". Assuming you are looking for an answer which must be a positive number (like the length of the side of a triangle), if it was "+" you put "+" in the middle of your answer. If it was "-" you put the a "-".

If there are two possible answers (a positive and a negative) then you switch signs, so if it the sign in Step 1 was "+", the other possible answer is "-√3 -√2/5". If it was "-", the other possible answer is "√2/5 -√3".

Here are the steps on the spreadsheet plus the look-up table. The symbols in columns B, D, F and H, i.e. √, (, ), +, - and =, are purely for decoration and are not used in the calculations:


Email me if you'd like the spreadsheet with workings :-)