Calculating 5th Roots - IN YOUR HEAD!

Multiplying a number by itself (e.g. $ 6 \times 6$, or $6^2 $) is relatively easy. Going back the other way - finding out what number has been squared to get a particular result (or finding the "square root", e.g. $ \sqrt{36} $) - is tougher: you need to either just know the answer or just guess it (and, of course, square it to see if you were right, and then if you're wrong have another guess, informed by the outcome of your first one).

Cubes (the cube of 6 would be written $ 6^3 $, or $6 \times 6 \times 6$, for example) are similarly relatively simple to calculate compared to cube roots (e.g. $ \sqrt[3]{216} $), which also require a bit of trial-and-improvement (if you're working with secondary-school level maths).

Raising numbers to the 5^th power (i.e. $ 6^5 $, or $6 \times 6 \times 6 \times 6 \times 6 \times 6$) is, again, fairly easy (though you'll probably need a piece of paper), but working out a 5^th root (so, figuring out that $ \sqrt[5]{7776} $ is actually 6) involves quite a bit of guessing, trying, failing, and guessing again.

Except... if you're trying to work out the 5^th root of a number that is the result of raising any two-digit number to the power of 5 there's a rather nifty trick that you can use to do it in your head. Indeed, with relatively little practise you'll be able to calculate 5^th roots quicker than someone who doesn't know the trick can bash it into a calculator.

I described this trick as part of a 24-hour maths marathon broadcast in October 2020. You can find out more about the event here, but if you'd like to find out how to calculate 5^th roots in your head really quickly, watch my section in the player below:

If you'd like me to add a readable description of the trick to this post, please let me know in the comments!