Question

If \( x = 1 + \sqrt[6]{2} + \sqrt[6]{4} + \sqrt[6]{8} + \sqrt[6]{16} + \sqrt[6]{32} \), then \( \left( 1 + \frac{1}{x} \right)^{24} \) is equal to:

• A. \(1 \)
• B. \(4 \)
• C. \(16 \)
• D. \(24 \)

Hint:

Recognizing geometric series can greatly simplify expressions involving sums of roots.

Answer 1

The Correct Option is C

Step 1: Simplify the expression for \( x \).
Let \( r = 2^{1/6} \). Then the expression for \( x \) can be written as a geometric series:
\[ x = 1 + r + r^2 + r^3 + r^4 + r^5 \]
Step 2: Use the formula for the sum of a geometric series.
The sum of a geometric series is given by \( S_n = a \cdot \frac{r^n - 1}{r - 1} \), where \( a = 1 \), \( r = 2^{1/6} \), and \( n = 6 \):
\[ x = \frac{(2^{1/6})^6 - 1}{2^{1/6} - 1} = \frac{2 - 1}{2^{1/6} - 1} = \frac{1}{2^{1/6} - 1} \]
Step 3: Find the value of \( 1 + \frac{1}{x} \).
First, compute \( \frac{1}{x} \):
\[ \frac{1}{x} = 2^{1/6} - 1 \]
Then:
\[ 1 + \frac{1}{x} = 1 + (2^{1/6} - 1) = 2^{1/6} \]
Step 4: Calculate \( \left( 1 + \frac{1}{x} \right)^{24} \).
\[ \left( 1 + \frac{1}{x} \right)^{24} = \left( 2^{1/6} \right)^{24} = 2^{(1/6) \cdot 24} = 2^4 \]
Step 5: Evaluate \( 2^4 \).
\[ 2^4 = 16 \]
Final Answer: \( \boxed{16} \)