# Mathematics Problem of the Week

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### Do this square root using pencil and paper.

$346846512-346846492$ Don't be put off by the telephone sized numbers and, if you do use a calculator, it's likely to be give an approximation.

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### Solution

Using DOTS we can rewrite the expression thus $(34684651+34684649)⁢(34684651-34684649)$ then do the additions ... $69369300×2$ now start factor decomposition ... $693693×100×2$ continue ... $77077×9×100×2$ $7007×11×9×100×2$ $1001×7×11×9×100×2$ and there we complete, perhaps, the obvious factorizations?
But checking though the divisibiliy rules we find that 1001 is a multiple of seven ... $143×49×11×9×100×2$ $13×49×121×9×100×2$ Rooting all the perfect squares ... $7×11×3×10⁢13×2$ So ... $346846512-346846492 = 2310⁢26$

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