It is quite easy to see that $0.0625^{-2.25} = 512$ by plugging this into a calculator.
Of course, mathematics existed for millennia before the invention of the calculator; is there a way to compute $0.0625^{-2.25}$ without resorting to calculator usage?
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0.0625^(-2.25) is $\frac{1}{16}$ raised to the $-\frac{9}{4}$. power. Now $\frac{1}{16}$ is $2^{-4}$, so $(2^{-4})^{-\frac{9}{4}}$ is the same as $2^{-4 \dot -\frac{9}{4}}$ which is $2^9$.
This site is a good reference for fractional exponentiation: http://www.purplemath.com/modules/exponent5.htm
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For these kinds of problems, it helps to be able to recognize decimal representation of common fractions. Or, if you don't know the fraction immediately, at least know that you should calculate what the fraction is.
If $0.25$ and $0.0625$ don't jump out at you as $1/4$ and $1/16$ then you might spend a little time acquainting yourself with some of these. Then, it's a matter of knowing rules for exponentiation, which the others have shown for you.
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A physicist would just try out these things with log tables and if he's old would use a slide rule, $$y = 0.0625^{-2.25}\\ \begin{align} \implies\log_{10}(y) &= -2.25\cdot\log(6.25 \times10^{-2})\\ &= -2.25 \cdot \left(\log_{10}(6.25) -2\right)\\ &\approx -2.25 \cdot(0.79588 -2)\\ &= 2.70927 \end{align}\\ \therefore\space y = 10^{2.70927} \approx 512 $$
$0.0625^{-2.25} = \frac{1}{16}^{-\frac{9}{4}} = 16^{\frac{9}{4}} = (2^4)^{\frac{9}{4}} = 2^9 = 512$