Question

Which is larger ${\left(2.5\right)}^6$ or ${\left(1.25\right)}^{12}$?

Answer 1

We write ${\left(2.5\right)}^6={\left(\frac{5}{2}\right)}^6=\frac{5^6}{2^6}$. Similarly, you may obtain
${\left(1.25\right)}^{12}={\left(\frac{5}{4}\right)}^{12}=\frac{5^{12}}{4^{12}}=\frac{5^{12}}{2^{24}}$
Thus you have to check which of the numbers $\frac{5^6}{2^6}$, $\frac{5^{12}}{2^{24}}$ is larger. Equivalently, you must determine the larger number between $5^6$ and $2^{18}$ (why?).
However, $5^2=25<32=2^5$. Hence
$5^6={\left(5^2\right)}^3<{\left(2^5\right)}^3=2^{15}<2^{18}$
Thus
$\frac{5^{12}}{2^{24}}=\frac{5^6}{2^{18}}\times \frac{5^6}{2^6}<\frac{5^6}{2^6}$
Hence ${\left(1.25\right)}^{12}<{\left(2.5\right)}^6$