De-nesting nested radicals

A "nested radical" is something like this "√(18 + 8√5)", in other words, the answer includes the square root of a (multiple or fraction of a) square root. I stumbled across this concept recently and have read a few articles since, I think they make it all too complicated.

You often end up with a nested radical when you are solving quadratic equations or trying to find the length of the hypotenuse.

For example, the adjacent and opposite of a right angle triangle are both "2 + √5" long (it's an isosceles right angle triangle i.e. a 45-45-90 triangle) - how long is the hypotenuse?

Using "first insides outsides last", (2 + √5) x (2 + √5) = 4 + 4√5 + 5
This simplifies to 9 + 4√5.
Multiply by 2 = 18 + 8√5.
The hypotenuse is the square root of that = √(18 + 8√5). Messy.

The steps to de-nesting this are as follows:
Start again with "18 + 8√5"
Change the "8" to "2", which means divide it by 4*.
Square the number you had to divide by = 16.
Multiply 5 (the number after the square root sign) by 16 = 80.
Restate it as "18 + 2√80"
Then see if you can find two numbers which add up to 18 and multiply to 80.
The answers are "8" and "10", obviously.**
So "√(18 + 8√5)" simplifies to "√8 + √10"***.

You can check this on a scientific calculator - either way, you end up with 5.9907.
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* There are endless wrinkles to this - what if you are trying to find the square root of "8 + √320"? You need a "2" in front of the square root sign, so just pop a "2" there anyway (multiply the implied "1" by "2") and divide 320 by 2 squared = 80.

Then there are multiples which are fractions or odd numbers; roots of fractions; negative numbers and so on. But they are all variations of the basic method.

** If you struggle with this step, and I often do, you can use the same approach as for solving quadratic equations.
18 ÷ 2 = 9.
(9 + u) x (9 - u) = 81.
81 - u^2 = 81.
U^2 = 1.
U = 1.
(9 + 1) x (9 - 1) = 80.

*** Don't forget the basic rule that the square root of 4 can be "2" or "-2". So although there's no such thing as a negative length of a hypotenuse, in a different context, "-√8 - √10" might also be a valid answer. If you get "√8 - √10" then the negative of that, "√10 - √8", is also a possible answer.
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Clearly, there are some nested radicals that you just can't simplify, unless you are happy ending up with an answer that includes a multiple of i (the square root of -1) which is sort of missing the point.

There is one handy shortcut to see whether you can solve it in the first place, which is to change the number before the square root sign to "1" and adjust the number after the square root sign accordingly.
If the whole number squared minus that number is itself a perfect square, there will be an answer.
For example, starting with "8 + √320" again:
18^2 = 324.
324 - 320 = 4.
"4" is a perfect square number, so there will be an answer.
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Right, that's a bit of fun with numbers for a Friday evening, clocking off now :-)