Calculating Square Roots - MPotW

Do this square root using pencil and paper.

$\sqrt{34684651^{2} - 34684649^{2}}$ 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 $\sqrt{(34684651 + 34684649) (34684651 - 34684649)}$ then do the additions ... $\sqrt{69369300 \times 2}$ now start factor decomposition ... $\sqrt{693693 \times 100 \times 2}$ continue ... $\sqrt{77077 \times 9 \times 100 \times 2}$ $\sqrt{7007 \times 11 \times 9 \times 100 \times 2}$ $\sqrt{1001 \times 7 \times 11 \times 9 \times 100 \times 2}$ and there we complete, perhaps, the obvious factorizations?
But checking though the divisibiliy rules we find that 1001 is a multiple of seven ... $\sqrt{143 \times 49 \times 11 \times 9 \times 100 \times 2}$ $\sqrt{13 \times 49 \times 121 \times 9 \times 100 \times 2}$ Rooting all the perfect squares ... $7 \times 11 \times 3 \times 10 \sqrt{13 \times 2}$ So ... $\sqrt{34684651^{2} - 34684649^{2}} = 2310 \sqrt{26}$

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Square Rooting

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Do this square root using pencil and paper.

Solution