It feels like an eternity since I last got my hands dirty with some proper math. So, here we are. I recently came across this radical in a question bank and piqued my interest.

How To Simplify This Radical — A whiteboard style expression that shows the following expression: √(√9 −√8) = ??
The radical expression — Illustration created by the author

The goal is to simplify this radical as much as possible. This task is nothing fancy, and becomes “radically” simple when you notice a hidden pattern. Consider that a hint.

Do you think you can simplify it? Give it a go, and all the best!

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Spoiler Alert

Beyond this section, I will be explicitly discussing solutions to this problem. If you plan on solving it on your own, I recommend pausing reading at this point.

You may continue reading and compare approaches once you are done with your attempt.

Setting Up The Radical Expression

Let’s go for the low-hanging fruit first, and simplify the inner terms of the radical to get the following simplified expression.

How To Simplify This Radical — A whiteboard style expression that shows the following expression: √(√9 −√8) = √(√(3*3) −√(2*2*2)) = √(3 −2√2)
Simplified expression —Math illustrated by the author

At this point, the astute reader might have already noted the pattern that I hinted at earlier. If not, the pattern that we are looking at here is none other than the binomial square formula.

How To Simplify This Radical — A whiteboard style illustration showing the following binomial square formula: (a−b)² = a² + b² − 2*a*b
The binomial square formula — Math illustrated by the author

Now, we only need to do a little manipulation of the current expression to start seeing the pattern we are looking for.

How To Simplify This Radical — A whiteboard style expression that shows the following expression: √(3 −2√2) = √(2 + 1 − 2√2)
Manipulated expression — Math illustrated by the author

Do you see the pattern yet? If not, worry not. We only need to take one more step to actually see it. In fact, that would also take us one step closer to the solution.

The Simplified Radical

All you need to do is multiply the right-most term with √1 to get it to the form we are looking for:

How To Simplify This Radical — A whiteboard style expression that shows the following expression: √(2 + 1 − 2√2) = √(2 + 1 − (2√2*√1)). This is of the form: (a−b)² = a² + b² − 2*a*b, where a=√2 and b=√1 OR a=√1 and b=√2. Therefore, the expression takes the form √+-(√2 − 1)²
Seeing the binomial square — Math illustrated by the author

Here, it is important to note there are two possibilities: (a = √2 and b = √1) or (a = √1 and b = √2). Consequently, our final solution will also have two possibilities.

How To Simplify This Radical — A whiteboard style illustration that shows the following solution: √+-(√2 − 1)² = +-(√2 − 1)^(2/2) = +-(√2 − 1)
The final solution — Math illustrated by the author

Once we express the radical expression as a binomial square, the outermost square root gets eliminated and we are left with the following solution. I believe that this is the simplest form of this radical expression.

Final Thoughts

Ifyou felt that this problem was too simple, worry not. It has been a long time since I last wrote about math problems, and I wanted to start with a simple problem.

With time, I aim to work on more challenging puzzles and problems. So, keep an eye out on this space for more interesting challenges in the future, and thanks for reading!

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Further reading that might interest you:

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Reference and credit: Simply Logical.

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