# How to do this approximation?

Question: Of the following, which is the best approximation of $$\sqrt{1.5}(266)^{3/2}$$

$$(A)~1,000~~~~(B)~2,700~~~~(C)~3,200~~~~(D)~4,100~~~~(E)~5,300$$

I used $1.5\approx1.44=1.2^2$ and $266\approx256=16^2$. Therefore the approximation by me is $4096$, so I chose $(D)$ which is wrong. The correct answer is $(E)$.

How should I find it out?

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Use Newtons Method. – Damien Aug 18 '11 at 3:05
This is the same question as this one. – Jack Aug 18 '11 at 3:25

$$\sqrt{1.5}\cdot266^{3/2}\approx1.2 \times 16^3 = 4915.2$$

The closest answer is (E) 5300. Great intuition on how to find simple approximations, but you forgot to multiply by $1.2$! Also note that $1.44<1.5$ and $256<266$, so you know the true answer must be above the discovered approximation, leaving only the last answer.

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Okay, this is embarrassing ... I did the calculation wrong. Sorry bothering you guys, this question is simpler than I thought. – Voldemort Aug 18 '11 at 3:42

$$\sqrt{\frac{3}{2}} \cdot ( \sqrt{266})^3 =\sqrt{\frac{3\cdot 266}{2}}\cdot (\sqrt{266})^2 = \sqrt{399} \cdot 266 \approx 266 \cdot 20 = 5320$$

This is closest to option (E)

Edit: Note that the only approximation I used here is $\sqrt{399}\approx \sqrt{400}$ so the result will differ by a factor of $\frac{\sqrt{399}}{20}$. This can be quickly approximated too, $\frac{\sqrt{399}}{20} = \sqrt{1-1/400} \approx 1 - 1/800 =0.9987$.

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Great solution! But I think my original answer is more intuitive. It is my calculation that went wrong, I forgot to multiply 1.2. – Voldemort Aug 18 '11 at 3:45
@Voldemort: I favor pulling all the squares out of the radical first, just as kuch nahi did. I see nothing unintuitive about it. – robjohn Aug 18 '11 at 4:21

$$\sqrt{1.5}(266)^{3/2} \approx \sqrt{\frac{16}{9}}(256)^{3/2} \approx \frac{4}{3} \times 4096 \approx 5460$$

Hence, $(E)$

N.B. kuch nahi's answer is probably the "right" one ; this seemed more intuitive to me since I didn't need to compute $16^3$.

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