Math: (Shortcut for) Simplifying (Denesting) Nested Square Roots

This is a senior high Algebra material.
If you're teaching (or tutoring) or probably still in high school, this method is kinda easy to understand. Kinda.

The basic form looks like this :

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Let's go to the methods and examples

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The plus thingy

√4 + √3 = 2 + √3

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The minus thingy

I forgot to put how to come up with 12 = 4 · 3 and 56 = 8 · 7.
The simplest way is to factor the number within the last square root. Then find the ones which if added will produce the first number.

For instance √(8 + 2√15)

We have 8 as the first number and 15 as the last one.
Factors of 15 :

• 1 · 15 = 15
• 3 · 5 = 15

Therefore, we choose the 2^nd one, the 3 · 5 = 15
because 3 + 5 = 8
So we have x = 3 and y = 5
We put all together : √(8 + 2√15) = √3 + √5

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Trick #1

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Trick #2

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If you wanna clarify the simplified forms above, you can go to Wolfram Alpha ➡ then type the complicated-er forms from my examples.

This is fun.