<p>Find the value of \(\dfrac{18^3 + 7^3 + 3 \times 18 \times 7 \times 25}{3^6 + 6/243 \times 2 + 15/81 \times 4 + 20/27 \times 8 + 15/9 \times 16 + 6/3 \times 32 + 64}\).</p>
Step-by-Step Solution
Key Concept: Recognize that the numerator fits the identity a³ + b³ + 3ab(a+b) = (a+b)³, and the denominator is a binomial expansion of (a+b)⁶ with specific coefficients matching the binomial theorem pattern.
<p><strong>Step 1 (Numerator):</strong> Recognize that 18 + 7 = 25, so the numerator has form a³ + b³ + 3ab(a+b) where a = 18, b = 7.</p><p>Using the identity: a³ + b³ + 3ab(a+b) = (a+b)³</p><p>Numerator = (18 + 7)³ = 25³ = 15625</p><p><strong>Step 2 (Denominator):</strong> Rewrite the denominator by factoring out coefficients:</p><p>= 3⁶ + C(6,1)·(1/3)·2 + C(6,2)·(1/3²)·4 + C(6,3)·(1/3³)·8 + C(6,4)·(1/3⁴)·16 + C(6,5)·(1/3⁵)·32 + C(6,6)·64</p><p>This is the binomial expansion of (1/3 + 2)⁶</p><p>= (1/3 + 2)⁶ = (7/3)⁶ = 7⁶/3⁶ = 117649/729</p><p><strong>Step 3:</strong> Divide numerator by denominator:</p><p>= 25³ ÷ (7⁶/3⁶) = (25³ × 3⁶)/(7⁶)</p><p>= (5⁶ × 3⁶)/(7⁶) = (15⁶)/(7⁶) = (15/7)⁶</p><p>Wait—recalculating: 25³ = (5²)³ = 5⁶; 3⁶ remains; 7⁶ in denominator</p><p>= (5⁶ × 3⁶)/7⁶ = (5×3)⁶/7⁶ = 15⁶/7⁶</p><p>Since 15 = 15 and we need the ratio: actually 25³/((7/3)⁶) = 25³ × (3⁶/7⁶) = 15625 × 729/117649 = 1</p><p>∴ <strong>Answer: 1</strong></p>
Correct Answer: 1