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- Feb 14, 2012
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Compute $\sqrt{2000(2007)(2008)(2015)+784}$ without the help of calculator.
Hey kaliprasad, thanks for participating and your method is good and I'm particularly very happy to see you finally picking up on LaTeX!Letting 2000 = a
we have 2000 * 2007 *2008 * 2015 + 784
a(a+7)(a+8)(a+15) + 784
= a(a+15) (a+7) (a+8) + 784
= (a^2+15a) (a^2 + 15a + 56) + 784
= $(a^2 + 15 a + 28 - 28 ) (a^2 + 15a + 28 + 28) + 28^2$
= $(a^2 + 15 a + 28)^2 - 28^2 + 28^2$
= $(a^2 + 15 a + 28)^2$
hence square root is $a^2 + 15a + 28$ or 4000000 + 30000 + 28
Nice one, agentmulder!Almost identical to MarkFL's , i just factored 16 to make the multiplications a bit smaller.
$ \sqrt{(2000)(2007)(2008)(2015)+ 784} \ = $
$ \sqrt{16 [ (\frac{2000}{4})(2007) ( \frac{2008}{4} ) (2015) + \frac{16 \cdot 7^2}{16} ]} \ = $
$ 4 \sqrt{(500)(2015)(502)(2007) \ + \ 7^2} \ = $
$ 4 \sqrt{(1007507 -7)(1007507 + 7) + 7^2} \ = $
$ 4 \sqrt{(1007507)^2} \ = \ 4030028 $
Admittedly , had i not seen MarkFL's method i probably would not have discovered it on my own.