Master the Nested Square Root Trick: A Simple Math Shortcut | ADRE Prep Notes

Sometimes we get a question like this: 'What is the value of √5√5√5√5√5?'

The answer comes out in a very easy trick: take the number, and raise it to the power of (2ⁿ - 1) / 2ⁿ where 'n' is the number of roots.

For this problem, the answer is 5^31/32.

But how do we get to this trick? Let's check out the concept.

First, you need to remember that taking a square root is the same as raising a number to the power of 1/2. So, √x = x^1/2.

We'll solve this by working from the inside out:

1. Look at the innermost number, √5. We can rewrite this as 5^1/2.

2. Now, let's take on the next root: √5√5. We've already figured out the inner part, so we can write this as: √(5 · 5^1/2) Use the rule of exponents to add the powers (1 + 1/2 = 3/2): √(5^3/2) = (5^3/2)^1/2 Now, multiply the exponents to get the result: 5^{(3/2 · 1/2)} = 5^3/4

3. Let's keep going. For √5√5√5, we'll use our previous result and a similar process: √(5 · 5^3/4) = √(5^7/4) = (5^7/4)^1/2 = 5^{(7/4 · 1/2)} = 5^7/8

4. You can see the pattern now. When you add the next root, you repeat the process: √(5 · 5^7/8) = √(5^15/8) = (5^15/8)^1/2 = 5^15/16

5. And for the final, fifth root: √(5 · 5^15/16) = √(5^31/16) = (5^31/16)^1/2 = 5^31/32

The Final Trick Revealed

You've just recreated the trick's underlying concept! You can see that for each square root you add, the exponent's denominator is multiplied by 2, and the numerator is always one less than the denominator.

The pattern looks like this:

• 1 root: 5^1/2
• 2 roots: 5^3/4
• 3 roots: 5^7/8
• 4 roots: 5^15/16
• 5 roots: 5^31/32

So, to solve this trick quickly, all you have to do is count the number of roots, let's call it 'n'. The answer is your base number to the power of (2ⁿ - 1) / 2ⁿ.

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