Without the use of a calculator , calculate :
$\displaystyle
\sqrt{1000\times1001\times1002\times1003}
$
Define
$\displaystyle z = x^3 \; (x^3 + 1) \; (x^3+2) \; (x^3+3)$
so
$\displaystyle z = [x^3 \; (x^3+3)] \; [(x^3+1) \; (x^3+2)]$
$\displaystyle z = (x^6 + 3x^3)(x^6 + 3x^3 +2)$
If we let $\displaystyle y = x^6 + 3x^3$
then
$\displaystyle z = y (y+2) = y^2 + 2y = (y+1)^2 -1$
so for large y,
$\displaystyle \sqrt{z} \approx y+1$
Now take $\displaystyle x = 10$, whence $\displaystyle y+1 = 10^6 + 3 \times 10^3 + 1$