Question

incorrect 1 try left. try once more estimate \\(\sqrt3{923}\\) to the nearest integer. \\(\sqrt3{923} \approx \\) \\(\underline{\quad}\\) 30

Explanation:

Step1: Find cubes around 923

We know that \(9^3 = 729\) and \(10^3=1000\).

Step2: Compare 923 with these cubes

Since \(729<923<1000\), we check the distance of 923 from \(9^3\) and \(10^3\). The distance from 923 to 729 is \(923 - 729=194\), and the distance from 1000 to 923 is \(1000 - 923 = 77\). Since 923 is closer to \(10^3 = 1000\)? Wait, no, wait, \(9^3=729\), \(10^3 = 1000\), but wait, \(9.7^3\approx912.67\), \(9.8^3=(9 + 0.8)^3=9^3+3\times9^2\times0.8 + 3\times9\times0.8^2+0.8^3=729+194.4 + 17.28+0.512 = 941.192\). Wait, 923 is between \(9.7^3\) and \(9.8^3\)? Wait, no, I made a mistake. Wait, \(9^3 = 729\), \(10^3=1000\), but wait, \(9^3=729\), \(10^3 = 1000\), but \(9.7^3=9.7\times9.7\times9.7 = 94.09\times9.7=912.673\), \(9.8^3=9.8\times9.8\times9.8 = 96.04\times9.8 = 941.192\), \(9.9^3=9.9\times9.9\times9.9=98.01\times9.9 = 970.299\). Wait, 923 is between \(9.7^3\) (912.673) and \(9.8^3\) (941.192). But we need to estimate to the nearest integer. Wait, but maybe I miscalculated the cubes of integers. Wait, \(9^3=729\), \(10^3 = 1000\), but \(9^3=729\), \(10^3=1000\), but what about \(9^3=729\), \(10^3=1000\), but \(9.7^3\approx912\), \(9.8^3\approx941\), so 923 is closer to 912 (which is \(9.7^3\)) or 941 (\(9.8^3\))? 923 - 912.673 = 10.327, 941.192 - 923 = 18.192. So closer to 9.7, but we need to the nearest integer. Wait, no, the question is to the nearest integer. Wait, \(9^3=729\), \(10^3=1000\), but wait, \(9^3=729\), \(10^3=1000\), but maybe I made a mistake in the initial step. Wait, no, the cube root of 923. Let's check \(9^3=729\), \(10^3=1000\), \(9^3 = 729\), \(10^3=1000\), but 923 is between \(9^3\) and \(10^3\). Now, the difference between 923 and 729 is 194, and between 1000 and 923 is 77. Wait, 77 is less than 194, so 923 is closer to 1000? But 1000 is \(10^3\), but 923 is 77 less than 1000, and 194 more than 729. So the nearest integer would be 10? Wait, no, that can't be. Wait, no, I think I messed up the cube roots. Wait, \(9^3=729\), \(10^3=1000\), \(9.7^3\approx912\), \(9.8^3\approx941\), so 923 is between 9.7 and 9.8, so the nearest integer is 10? Wait, no, 9.7 is closer to 10? No, 9.7 is closer to 10? Wait, 9.7 is 0.3 away from 10, and 0.7 away from 9. So no, 9.7 is closer to 10? Wait, no, 9.7 is 0.3 from 10, 0.7 from 9. So 9.7 is closer to 10. But 923 is 9.7^3≈912.67, 923 - 912.67 = 10.33, and 941 - 923 = 18.19, so 923 is closer to 9.7^3, which is closer to 10? Wait, no, 9.7 is closer to 10, but the cube root of 923. Wait, maybe I made a mistake earlier. Wait, let's calculate \(9^3 = 729\), \(10^3 = 1000\), \(9^3=729\), \(10^3=1000\), the number 923 is between 9^3 and 10^3. The midpoint between 729 and 1000 is (729 + 1000)/2 = 864.5. Since 923 > 864.5, it's closer to 1000, so the nearest integer is 10? But that contradicts the earlier calculation. Wait, no, wait, \(9.7^3=912.67\), \(9.8^3=941.19\), so 923 is between 9.7 and 9.8, so the nearest integer is 10? Wait, 9.7 is 9.7, so the nearest integer is 10? Wait, no, 9.7 is closer to 10? Yes, because 9.7 is 0.3 away from 10 and 0.7 away from 9. So the nearest integer is 10. Wait, but let's check \(9^3=729\), \(10^3=1000\), 923 is 77 less than 1000, and 194 more than 729. So the distance to 1000 is smaller, so the nearest integer is 10.

Wait, but maybe I made a mistake. Let's check \(9^3=729\), \(10^3=1000\), \(9.7^3≈912\), \(9.8^3≈941\), so 923 is 923 - 912.67 = 10.33 above 9.7^3, and 941.19 - 923 = 18.19 below 9.8^3. So it's closer to 9.7^3, which is 9.7, so the nearest integer is 10? Wait, 9.7 rounded to the nearest integer is 10? No, 9.7 rounded to the nearest integer is 10? Wait, 9.5 and above rounds up to 10, so 9.7 rounds to 10. So the nearest integer is 10.

Wait, but the initial wrong answer was 30, which is way off. So the correct approach is to find two perfect cubes that 923 lies between. Let's find \(n^3 \leq 923 < (n + 1)^3\). Let's try n=9: \(9^3=729\), n=10: \(10^3=1000\). So 729 ≤ 923 < 1000. Now, calculate the distance from 923 to 729: 923 - 729 = 194. Distance from 1000 to 923: 1000 - 923 = 77. Since 77 < 194, 923 is closer to 1000, so the nearest integer is 10.

Answer:

10