# Without the use of a calculator

• November 14th 2009, 01:06 AM
dhiab
Without the use of a calculator
Without the use of a calculator , calculate :
$
\sqrt{1000\times1001\times1002\times1003}
$
• November 14th 2009, 01:22 AM
Bacterius
What do you mean by "calculating" ? You mean evaluate, because $1000 \times 1001 \times 1002 \times 1003$ is definitely not a perfect square (it is nearly one, but it's not) ... so we can only approach it !
• November 14th 2009, 11:28 AM
awkward
Quote:

Originally Posted by dhiab
Without the use of a calculator , calculate :
$
\sqrt{1000\times1001\times1002\times1003}
$

Define
$z = x^3 \; (x^3 + 1) \; (x^3+2) \; (x^3+3)$
so
$z = [x^3 \; (x^3+3)] \; [(x^3+1) \; (x^3+2)]$
$z = (x^6 + 3x^3)(x^6 + 3x^3 +2)$

If we let $y = x^6 + 3x^3$
then
$z = y (y+2) = y^2 + 2y = (y+1)^2 -1$
so for large y,
$\sqrt{z} \approx y+1$

Now take $x = 10$, whence $y+1 = 10^6 + 3 \times 10^3 + 1$