Answering a question containing $\sqrt{1.5}$ without using calculator. Of the following which is the best approximation of $\sqrt{1.5}(266)^{\frac{3}{2}}$?
(A)1,000
(B)2,700
(C)3,200
(D)4,100
(E)5,300
How can I answer this without using a calculator and in about 2.5 minutes?

$\sqrt{1.5}(266)^{\frac{3}{2}} 
   = 266 \times\sqrt{1.5 \times 266}
   = 266 \times\sqrt{266 + 133}
   = 266 \times\sqrt{399}$
Note that $\sqrt{399}$ is very close to $\sqrt{400} = 20$.
So $266 \times\sqrt{399} \approx 266 \times 20 = 5320.$

By the way, if you want a bit more accurracy, let $f(x) = \sqrt{x}$. Then 
$f'(x) = \dfrac{1}{2 \sqrt x}$
\begin{align}
   f(400+\delta) &\approx f(400) + \delta f'(400) \\
   f(400-1) &\approx \sqrt{400} - \dfrac{1}{2 \sqrt{400}} \\
   \sqrt{399} &\approx 20 - \dfrac{1}{40} \\
\hline
   266 \times\sqrt{399} 
   &\approx 266 \times \left(20-\dfrac{1}{40} \right) \\
   &\approx 5320 - 6.65 \\
   &\approx 5313.35
\end{align}
To ten digits, the correct answer is $5313.345839$.
