Question

which number is closest to \\(\sqrt{108}\\) ? 10.3 10.8 9.9 10.7

Explanation:

Step1: Calculate square of 10.3

$10.3^2 = 10.3\times10.3 = 106.09$

Step2: Calculate square of 10.8

$10.8^2 = 10.8\times10.8 = 116.64$

Step3: Calculate square of 9.9

$9.9^2 = 9.9\times9.9 = 98.01$

Step4: Calculate square of 10.7

$10.7^2 = 10.7\times10.7 = 114.49$

Step5: Compare differences from 108

• Difference for 10.3: $|108 - 106.09| = 1.91$
• Difference for 10.8: $|116.64 - 108| = 8.64$
• Difference for 9.9: $|108 - 98.01| = 9.99$
• Difference for 10.7: $|114.49 - 108| = 6.49$

The smallest difference is 1.91 for 10.3? Wait, no, wait, actually we can also calculate $\sqrt{108}\approx10.392$. Now find the difference between 10.392 and each option:

• $|10.392 - 10.3| = 0.092$
• $|10.392 - 10.8| = 0.408$
• $|10.392 - 9.9| = 0.492$
• $|10.392 - 10.7| = 0.308$

So the smallest difference is with 10.3? Wait, no, wait my first approach was wrong. Let's recalculate. Wait, $\sqrt{108}=\sqrt{36\times3}=6\sqrt{3}\approx6\times1.732 = 10.392$. Now compare with each option:

• 10.3: $10.392 - 10.3 = 0.092$
• 10.8: $10.8 - 10.392 = 0.408$
• 9.9: $10.392 - 9.9 = 0.492$
• 10.7: $10.7 - 10.392 = 0.308$

So the closest is 10.3? Wait, but wait, maybe I made a mistake earlier. Wait, 10.3 squared is 106.09, 10.4 squared is 108.16. Oh! Wait, 10.4 squared is 108.16, which is very close to 108. But the options don't have 10.4. Wait the options are 10.3, 10.8, 9.9, 10.7. So 10.3 squared is 106.09, 10.4 squared is 108.16. So $\sqrt{108}$ is between 10.3 and 10.4. So the distance from 10.3: 10.392 - 10.3 = 0.092. From 10.4 (but not an option). Among the given options, 10.3 is the closest? Wait no, wait 10.392 - 10.3 = 0.092, 10.4 - 10.392 = 0.008, but 10.4 is not an option. So among the given options, 10.3 is the closest? Wait no, wait 10.3 squared is 106.09, 10.392 squared is 108. So the difference between 10.3 and 10.392 is 0.092, between 10.7 and 10.392 is 0.308, between 10.8 and 10.392 is 0.408, between 9.9 and 10.392 is 0.492. So yes, 10.3 is the closest. Wait but wait, maybe I miscalculated 10.3 squared. Let's recalculate 10.310.3: 1010=100, 100.3=3, 0.310=3, 0.30.3=0.09, so (10+0.3)^2=10^2 + 210*0.3 + 0.3^2=100+6+0.09=106.09. Correct. 10.4^2=108.16. So 108 is 108.16 - 0.16, so sqrt(108) is 10.4 - a little bit. So the closest among the options is 10.3? Wait no, 10.392 is closer to 10.4, but since 10.4 is not an option, the next closest is 10.3 (difference 0.092) and 10.7 (difference 0.308), 10.8 (0.408), 9.9 (0.492). So 10.3 is the closest.

Answer:

10.3