<p>The integral part of \((8 + 3\sqrt{7})^{20}\) is even. (State whether true or false.)</p>
Step-by-Step Solution
Key Concept: Use binomial theorem to express (8 + 3√7)^20 + (8 - 3√7)^20, which is always an integer. The integral part of (8 + 3√7)^20 equals this sum minus 1, since 0 < (8 - 3√7)^20 < 1.
<p><strong>Step 1:</strong> Note that 8 - 3√7 > 0 since (3√7)² = 63 < 64. Also, 8 - 3√7 ≈ 8 - 7.937 ≈ 0.063, so 0 < (8 - 3√7)^20 < 1.</p><p><strong>Step 2:</strong> By binomial theorem, (8 + 3√7)^20 + (8 - 3√7)^20 = 2∑(even terms) = an even integer N.</p><p><strong>Step 3:</strong> Therefore: (8 + 3√7)^20 = N - (8 - 3√7)^20</p><p><strong>Step 4:</strong> Since 0 < (8 - 3√7)^20 < 1, the integral part of (8 + 3√7)^20 is N - 1.</p><p><strong>Step 5:</strong> Since N is even, N - 1 is odd.</p><p>∴ Answer: <strong>False</strong> (The integral part is odd, not even)</p>
Correct Answer: A