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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The Duplex Method for Squaring Numbers

The Duplex method is a fast and efficient way to calculate the squares of numbers, a technique derived from Vedic Mathematics. At its core is the concept of a "duplex," which is a specific calculation performed on a digit or group of digits.

The Duplex Rules

• Duplex of a single digit (a): D(a) = a²
• Duplex of two digits (ab): D(ab) = 2 × a × b
• Duplex of three digits (abc): D(abc) = 2 × a × c + b²

The General Process

The process for squaring a number using the Duplex method is consistent, regardless of the number of digits. Follow these steps:

1. Determine the "Duplex Columns": The number of columns and the duplexes to be calculated follow a mirrored pattern, starting from a single digit on the right and expanding outwards.

   • For a 2-digit number (ab): The columns are D(a) / D(ab) / D(b)
   • For a 3-digit number (abc): The columns are D(a) / D(ab) / D(abc) / D(bc) / D(c)

2. Calculate the Duplex for Each Column: Go through each column and calculate its value using the rules above.

3. Combine the Results with Carries (from right to left):

   • Start with the rightmost column. Write down the last digit of the result.
   • Carry over any remaining digits to the next column on the left.
   • Add the carry-over to the value of that next duplex column.
   • Repeat this process until you reach the leftmost column. The final sum of this column, including any carry-over, forms the first digits of your answer.

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Example: Squaring a 2-Digit Number (47)

Number: ab = 47

Duplex Columns: D(4) / D(47) / D(7)

1. Calculate Duplexes:

D(4) = 4² = 16
D(47) = 2 × 4 × 7 = 56
D(7) = 7² = 49

2. Combine with Carries (Right to Left):

• Column 3 (D(7)): The value is 49. Write down 9, carry over 4.
• Column 2 (D(47)): The value is 56. Add the carry: 56 + 4 = 60. Write down 0, carry over 6.
• Column 1 (D(4)): The value is 16. Add the carry: 16 + 6 = 22. Write down 22.

Final Answer: 2209

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Example: Squaring a 3-Digit Number (147)

Number: abc = 147

Duplex Columns: D(1) / D(14) / D(147) / D(47) / D(7)

1. Calculate Duplexes:

D(1) = 1² = 1
D(14) = 2 × 1 × 4 = 8
D(147) = 2 × 1 × 7 + 4² = 14 + 16 = 30
D(47) = 2 × 4 × 7 = 56
D(7) = 7² = 49

2. Combine with Carries (Right to Left):

• Column 5 (D(7)): The value is 49. Write down 9, carry over 4.
• Column 4 (D(47)): The value is 56. Add the carry: 56 + 4 = 60. Write down 0, carry over 6.
• Column 3 (D(147)): The value is 30. Add the carry: 30 + 6 = 36. Write down 6, carry over 3.
• Column 2 (D(14)): The value is 8. Add the carry: 8 + 3 = 11. Write down 1, carry over 1.
• Column 1 (D(1)): The value is 1. Add the carry: 1 + 1 = 2. Write down 2.

Final Answer: 21609

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