Question

which has a value closest to 9? \\(\sqrt{83}\\) \\(\sqrt{63}\\) \\(\sqrt{99}\\) \\(\sqrt{10}\\)

Explanation:

Step1: Recall that \( \sqrt{x^2} = x \), so we find the squares around 9. \( 9^2 = 81 \), \( 10^2 = 100 \).

Step2: Compare each radicand to 81 and 100.

• For \( \sqrt{83} \): \( 83 - 81 = 2 \)
• For \( \sqrt{63} \): \( 81 - 63 = 18 \)
• For \( \sqrt{99} \): \( 100 - 99 = 1 \) (Wait, no: \( 99 - 81 = 18 \)? Wait, no, we compare to \( 9^2 = 81 \) and \( 10^2 = 100 \). Wait, \( 9^2 = 81 \), so \( \sqrt{81} = 9 \). So we find the distance of each radicand from 81.
• \( |83 - 81| = 2 \)
• \( |63 - 81| = 18 \)
• \( |99 - 81| = 18 \) (Wait, no, \( 99 - 81 = 18 \), but \( 100 - 99 = 1 \), but we want distance from 81 (since \( \sqrt{81}=9 \)). Wait, actually, the value of \( \sqrt{x} \) is closest to 9 when \( x \) is closest to \( 9^2 = 81 \).

So:

• \( 83 - 81 = 2 \)
• \( 81 - 63 = 18 \)
• \( 99 - 81 = 18 \)
• \( 81 - 10 = 71 \)

So \( \sqrt{83} \) has radicand 83, which is 2 away from 81, so \( \sqrt{83} \) is closest to 9. Wait, but wait, \( \sqrt{99} \) is \( \sqrt{100 - 1} \approx 10 - \frac{1}{20} \) (using linear approximation), but actually, the difference from 81: 99 - 81 = 18, while 83 - 81 = 2. So 83 is closer to 81 than 99 is? Wait, no, 99 is closer to 100 (which is \( 10^2 \)), but we want closer to 9 (which is \( \sqrt{81} \)). So the distance from 81: 83 is 2 away, 99 is 18 away, 63 is 18 away, 10 is 71 away. So \( \sqrt{83} \) is closest. Wait, but wait, \( \sqrt{99} \) is \( \approx 9.95 \), which is 0.95 away from 9. \( \sqrt{83} \approx 9.11 \), which is 0.11 away from 9. Oh! Wait, I made a mistake earlier. I should calculate the actual value.
Let's calculate each:

• \( \sqrt{83} \approx 9.11 \) (since \( 9^2 = 81 \), \( 9.1^2 = 82.81 \), \( 9.11^2 \approx 83 \))
• \( \sqrt{63} \approx 7.94 \)
• \( \sqrt{99} \approx 9.95 \)
• \( \sqrt{10} \approx 3.16 \)

Now, find the absolute difference from 9:

• \( |9.11 - 9| = 0.11 \)
• \( |7.94 - 9| = 1.06 \)
• \( |9.95 - 9| = 0.95 \)
• \( |3.16 - 9| = 5.84 \)

So comparing 0.11, 1.06, 0.95, 5.84: the smallest is 0.11 (from \( \sqrt{83} \)) and 0.95 (from \( \sqrt{99} \)). Wait, 0.11 is smaller than 0.95. So \( \sqrt{83} \) is closer. Wait, but earlier when I thought \( 99 - 81 = 18 \), but actually, the value of \( \sqrt{99} \) is about 9.95, which is 0.95 away from 9, while \( \sqrt{83} \) is about 9.11, 0.11 away. So \( \sqrt{83} \) is closer. Wait, but let's check the radicand distance again. \( 83 - 81 = 2 \), \( 100 - 99 = 1 \), but \( \sqrt{99} \) is \( \sqrt{100 - 1} \approx 10 - \frac{1}{20} = 9.95 \), so distance from 9 is 0.95. \( \sqrt{83} \) is \( \sqrt{81 + 2} \approx 9 + \frac{2}{18} \approx 9.11 \) (using linear approximation \( \sqrt{a + b} \approx \sqrt{a} + \frac{b}{2\sqrt{a}} \) for small b). So \( \frac{2}{2*9} = \frac{1}{9} \approx 0.11 \), so distance from 9 is ~0.11. So \( \sqrt{83} \) is closer.

Answer:

\(\boldsymbol{\sqrt{83}}\)