(2^2)/3 x (3^2)/8 x (4^2)/15 x ... x (9^2)/80 (2^2)/3 x (3^2)/8 x (4^2)/15 x ... x (9^2)/80
Tham khảo câu trả lời từ Trợ lý ảo Lazi To simplify and calculate the expression \[ \left( \frac{2^2}{3} \right) \times \left( \frac{3^2}{8} \right) \times \left( \frac{4^2}{15} \right) \times \left( \frac{5^2}{24} \right) \times \left( \frac{6^2}{35} \right) \times \left( \frac{7^2}{48} \right) \times \left( \frac{8^2}{63} \right) \times \left( \frac{9^2}{80} \right) \] we begin by identifying the pattern in the denominator. The denominators are as follows: - \(3 = 1 \cdot 3\) - \(8 = 2^3\) - \(15 = 3 \cdot 5\) - \(24 = 4! = 4 \cdot 6\) - \(35 = 5 \cdot 7\) - \(48 = 6 \cdot 8\) - \(63 = 7 \cdot 9\) - \(80 = 8 \cdot 10\) Now let's denote this product as \( P \): \[ P = \frac{2^2 \cdot 3^2 \cdot 4^2 \cdot 5^2 \cdot 6^2 \cdot 7^2 \cdot 8^2 \cdot 9^2}{3 \cdot 8 \cdot 15 \cdot 24 \cdot 35 \cdot 48 \cdot 63 \cdot 80} \] We first calculate the numerator: \[ 2^2 \cdot 3^2 \cdot 4^2 \cdot 5^2 \cdot 6^2 \cdot 7^2 \cdot 8^2 \cdot 9^2 = (2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9)^2 \] Now calculating \( 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \), which is \( 9! \): \[ 9! = 362880 \] So the numerator becomes: \[ (9!)^2 = (362880)^2 \] Now calculate the denominator: 1. **Compute the denominators explicitly**: - \( 3 = 3 \) - \( 8 = 2^3 = 8 \) - \( 15 = 3 \cdot 5 = 15 \) - \( 24 = 4! = 24 \) - \( 35 = 5 \cdot 7 = 35 \) - \( 48 = 6 \cdot 8 = 48 \) - \( 63 = 7 \cdot 9 = 63 \) - \( 80 = 8 \cdot 10 = 80 \) Calculating the product of the denominators: \[ 3 \cdot 8 \cdot 15 \cdot 24 \cdot 35 \cdot 48 \cdot 63 \cdot 80 \] Calculating step by step: - \(3 \cdot 8 = 24\) - \(24 \cdot 15 = 360\) - \(360 \cdot 24 = 8640\) - \(8640 \cdot 35 = 302400\) - \(302400 \cdot 48 = 14515200\) - \(14515200 \cdot 63 = 915840000\) - \(915840000 \cdot 80 = 73267200000\) Thus, the denominator is \( 73267200000 \). Finally, we can compute the entire expression \( P = \frac{(9!)^2}{3 \cdot 8 \cdot 15 \cdot 24 \cdot 35 \cdot 48 \cdot 63 \cdot 80} \): \[ P = \frac{(362880)^2}{73267200000} \] Calculating \( (362880)^2 = 131681894400 \). Thus, \[ P = \frac{131681894400}{73267200000} \approx 1.795 (approx) \] After final simplification, if you want an exact fraction, we would reduce However, it ends quite close to 1.8. Thus, the final answer is: \[ P \approx 1.8 \] So the final boxed answer is: \[ \boxed{1.8} \]