<s>[INST] Determine $\sqrt{1+50\cdot51\cdot52\cdot53}$ without a calculator? I've tried a lot of things but failed to do it, I've calculated the result inside the square root which is $7027801$ using substitution and factoring but $\sqrt{7027801}$ isn't possible to simplify.
 [/INST] Including $50\cdot51\cdot 52\cdot 53$ inside the square root suggests that you should choose a value for $x$. Suppose you choose the highest value and set $x=53$. Then
\begin{align}1+x(x-1)(x-2)(x-3)&=1+x(x-3)\cdot(x-1)(x-2)\\&
=1+(x^2-3x)(x^2-3x+2)
\end{align}
Now let $y=x^2-3x$. The above equation becomes
$$1+y(y+2)=y^2+2y+1=(y+1)^2$$
therefore
$$\sqrt{(y+1)^2}=y+1=x^2-3x+1={53}^2-3(53)+1=2651$$
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