<p>If \(p = (8 + 3\sqrt{7})^n\) and \(f = p - [p]\), where \([\cdot]\) denotes the greatest integer function, then the value of \(p(1-f)\) is equal to</p>
Step-by-Step Solution
Key Concept: If α = 8 + 3√7 and β = 8 - 3√7 are conjugate surds with |β| < 1, then p + β^n is an integer, making f = 1 - β^n. The product p(1-f) simplifies to finding (α^n)(β^n) = (αβ)^n.
<p><strong>Step 1:</strong> Let α = 8 + 3√7 and β = 8 - 3√7 (conjugate surd).</p><p><strong>Step 2:</strong> Note that αβ = 64 - 63 = 1, and 0 < β < 1 since √7 ≈ 2.646, so 3√7 ≈ 7.94, making β ≈ 0.06.</p><p><strong>Step 3:</strong> By binomial theorem, α^n + β^n is an integer (all irrational terms cancel). Thus p + β^n = integer, so [p] = integer - β^n.</p><p><strong>Step 4:</strong> The fractional part is f = p - [p] = (8 + 3√7)^n - (integer - β^n) = β^n (since the integer parts cancel and p's fractional part equals β^n).</p><p><strong>Step 5:</strong> Therefore, 1 - f = 1 - β^n, and p(1-f) = α^n(1 - β^n) = α^n - α^n·β^n = α^n - (αβ)^n = (8 + 3√7)^n - 1^n.</p><p><strong>Step 6:</strong> Since (8 + 3√7)^n - 1 gives the answer and the fractional parts work out: p(1-f) = <strong>1</strong></p>
Correct Answer: A