No Calculator Needed (2)

Algebra Level 3

$\large \sqrt{(8000\times 8006\times 8008\times 8014)+ 576} = \ ?$

If you did use a calculator, good job! You have gained some points, but you didn't learn anything. If you didn't, good job! You are getting better at math.

The answer is 64112024.

2 solutions

Uahnbu Tran
Jun 4, 2015

$Let\quad x\quad be\quad 8007\\ The\quad expression\quad =\quad \sqrt { (x-7)(x-1)(x+1)(x+7)+576 } \\ \qquad =\sqrt { { (x }^{ 2 }-49){ (x }^{ 2 }-1)+576 } \\ \qquad =\sqrt { { x }^{ 4 }-50{ x }^{ 2 }+49+576 } \\ \qquad =\sqrt { { ({ x }^{ 2 }-25) }^{ 2 } } \\ \qquad =\left| { x }^{ 2 }-25 \right|$

Of course haha

UahNBu Tran - 8 months, 1 week ago

And you can do 8007 squared without a calculator?

Casey Hubbell - 6 years ago

Of course we can. Use a pen, and a paper: $\quad \quad \quad \quad \quad \quad 8007\\ \cfrac { \quad \quad \quad \quad \times \quad 8007 }{ \quad \quad \quad \quad \quad 56049\\ \cfrac { +\quad 64056 }{ \quad \quad \quad 64112049 } }$

UahNBu Tran - 5 years, 11 months ago

Rameswari Kubendran
Jun 4, 2015

Let us take 8000 as ‘a’ Then 8000 x 8006 x 8008 x 8014 can be written as a(a+6)(a+8)(a+14) Grouping them as a(a+14)(a+8)(a+6) = (a2+14a) (a2+14a+48) Now let as take (a2+14a) as x then (a2+14a) (a2+14a+48) becomes x(x+48) (8000 x 8006 x 8008 x 8014 )+576= x2+48x+576 = (x+24)2 Sqrt(8000 x 8006 x 8008 x 8014 )+576= x2+48x+576)= x+24 = a2+14a+24 = 80002+8000(14)+24 = 64000000+ 112000+24 = 64112024

No Calculator Needed (2)

If you did use a calculator, good job! You have gained some points, but you didn't learn anything. If you didn't, good job! You are getting better at math.

The answer is 64112024.

2 solutions

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