# Without the use of a calculator

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

Originally Posted by dhiab
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 $\displaystyle y+1 = 10^6 + 3 \times 10^3 + 1$