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\[\large{log_{\sqrt2}{x} + log_{\sqrt{\sqrt2}}{x} + log_{\sqrt{\sqrt{\sqrt2}}}{x} + log_{\sqrt{\sqrt{\sqrt{\sqrt2}}}}{x}+ \text{upto 7 terms} = 1016}\]

Find the value of \(x\).

Find the minimum value of the expression below for real \(x\).

\[x^2-12x+40\]

If \(x\) and \(y\) are real numbers such that \(\dfrac{x^3+3y^3}{x^3-3y^3} = \dfrac{212}{131}\), what is the value of \(\dfrac{x+2y}{x-2y}\)?

\[\large \left(1+x^{\dfrac{1}{2}}\right)\left(1+x^{\dfrac{1}{4}}\right)\left(1+x^{\dfrac{1}{8}}\right)\left(1+x^{\dfrac{1}{16}}\right)\left(1-x^{\dfrac{1}{16}}\right) = ?\]

If \(m=a^x\), \(n=a^y\) and \(a^2=(m^yn^x)^z\), find the value of \(xyz\).

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