Without the help of calculator, evaluate $\sqrt[8]{10828567056280801}$.

Without the help of calculator, evaluate $\sqrt[8]{10828567056280801}$.

This might look like there will be a lot of guessing before getting the right answer. But, if you toy around with the figure $10828567056280801$, it's not hard to see we could rewrite it so that we have:

$10828567056280801$

$=(1\times 10^{16})+(8\times 10^{14})+(28\times 10^{12})+(56\times 10^{10})+(70\times 10^{8})+(56\times 10^{6})+(28\times 10^{4})+(8\times 10^{2})+(1)$

Can you see it now the very clear pattern that we have and made our conclusion to it?

Yeap! First, we noticed the coefficient of the power of $10$ are symmetric, then we suspect the given figure can be related to binomial expansion formula

$(a+b)^8=a^8+8 a^7 b+28 a^6 b^2+56 a^5 b^3+70 a^4 b^4+56 a^3 b^5+28 a^2 b^6+8 a b^7+b^8$

Therefore,

$(100+1)^8$

$=(10^2+1)^8$

$=(10^2)^8+8 (10^2)^7 (1)+28 (10^2)^6 (1)^2+56 (10^2)^5 (1)^3+70 (10^2)^4 (1)^4+56 (10^2)^3 (1)^5+28 (10^2)^2 (1)^6+8 (10^2) (1)^7+(1)^8$

$=(1\times 10^{16})+(8\times 10^{14})+(28\times 10^{12})+(56\times 10^{10})+(70\times 10^{8})+(56\times 10^{6})+(28\times 10^{4})+(8\times 10^{2})+(1)$

At this point, we can safely express $\sqrt[8]{10828567056280801}=\sqrt[8]{(100+1)^8}=101$ and we're hence done.