SAT Estimation

Correct Answer: C

Solution 1:

Tip: Recognize first few perfect squares $(1, 4, 9, ... 400)$ and cubes $(1, 8, 27,... 1000).$
If $x=\sqrt{108},$ then $x^2 = 108.$ We know that $10^2 = 100$ and $11^2 = 121.$ Since 108 is closer to 100 than it is to 121, then $x$ is probably closer to 10 than it is to 11. But let us check, because as Pi Han shows us, this is not always true!

We know that if $a^2 < x^2 < b^2,$ then $|a| < |x| < |b|.$ Let $a=10, x=\sqrt{108}$ and $b=10.5.$ Is the inequality $10^2 < (\sqrt{108})^2 < 10.5^2$ true? It is, since $100 < 108 < 110.25.$ So, we can conclude that $10 < \sqrt{108} < 10.5 < 11,$ which means that $\sqrt{108}$ is closer to 10 than it is to 11.Therefore, the answer is (C).

Solution 2:

Tip: Use a calculator.

$\sqrt{108}=10.392 \approx 10.$

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Incorrect Choices:

(A) , (B) , (D) , and (E)
Tip: Recognize first few perfect squares $(1, 4, 9, ... 400)$ and cubes $(1, 8, 27,... 1000).$
$8^2 = 64, 9^2=81, 11^2 = 121,$ and $12^2 = 144.$
None of these choices is closer to 108 than choice (C).

If you got this problem wrong, you should review SAT Numbers .

We can apply the use of AM GM property: with numbers $10,11$ , because they are distinct, the inequality doesn't hold. So,

$10.5 = \frac {10 + 11}{2} > \sqrt{10 \cdot 11} = \sqrt{110} > \sqrt{108}$

Which means that $\sqrt{108}$ is closer to $10$ than it is to $11$ . Hence the answer is $\boxed{10}$ .

I solved it by taking root of 100 since its close to 108. Square root of 100 is 10 so estimated value for square root of 108 will be 10