Answering a question containing $\sqrt{1.5}$ without using calculator.

Question

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?

Answer 1

$$\sqrt{\frac{3}{2} \times 266^3} = \sqrt{3 \times 133 \times 266^2} = 266 \times \sqrt{399} \simeq 266 \times 20 = 5320$$

The difference between $5320$ and the true answer is no more than $266$, since $\sqrt{399}$ is no more than $1$ away from $400$. (It's actually a much better approximation even than that.)

Answer 2

The given 4 alternatives are far from each other, and so we can calculate approximately choosing a convenient number close by.

(1) To calculate $266^{3/2}$, square root of 266 is needed. We approximate 266 to 256 as its square roots is nice $16$. Now to compensate for the reduction of this value we will increase $\sqrt{150/100}$ to $\sqrt{169/100}=1.3$.

So the final answer will be close to $1.3\times 16^3=1.3\times4096$ which should exceed 5000. So I will vote for the option E) $5300$