MHB Compute a square root of a sum of two numbers

AI Thread Summary
To compute $\sqrt{2000(2007)(2008)(2015)+784}$ without a calculator, the expression can be rearranged to fit the form $(a+b)(a-b)$, where $b^2=784$. This clever manipulation highlights the relationship between the four numbers involved. Participants praised the solution and noted improvements in LaTeX usage among contributors. The discussion emphasizes the importance of recognizing patterns in mathematical problems for simplification.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Compute $\sqrt{2000(2007)(2008)(2015)+784}$ without the help of calculator.
 
Mathematics news on Phys.org
My solution:

$$2000\cdot2015=4030028-28$$

$$2007\cdot2008=4030028+28$$

$$784=28^2$$

Hence:

$$2000\cdot2007\cdot2008\cdot2015+784=4030028^2-28^2+28^2=4030028^2$$

And so:

$$\sqrt{2000\cdot2007\cdot2008\cdot2015+784}=\sqrt{4030028^2}=4030028$$
 
This is probably the most genius way to collect the four numbers $2000$, $2007$, $2008$ and $2015$ in such a manner so that their product takes the form $(a+b)(a-b)$ and what's more, $b^2=784$!

Bravo, MarkFL!(Clapping)(Sun)
 
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
 
kaliprasad said:
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
Hey kaliprasad, thanks for participating and your method is good and I'm particularly very happy to see you finally picking up on LaTeX!(Tongueout)(Sun)
 
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.

:)
 
Last edited:
agentmulder said:
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.

:)

Nice one, agentmulder!:)(Sun)
 
Back
Top