Question

Which is larger: ${0.25}^4$ or ${0.35}^3$

Answer 1

We use the simple observation: If $a$ and $b$ are non zero numbers then $a<b$ its equivalent to $\frac{a}{b}<1$.
Write $0.25=\frac{1}{4}$ and $0.35=\frac{35}{100}=\frac{7}{20}$. We have to compare $\frac{1}{4^4}$ and $\frac{7^3}{{20}^3}$. But ${20}^3={(4\times 5)}^3=4^3\times 5^3$. Thus we see that
$\frac{{\left(0.25\right)}^4}{{\left(0.35\right)}^3}=\frac{4^3\times 5^3}{4^4\times 7^3}=\frac{5^3}{4\times 7^3}<1$
since $5<7$. Thus ${\left(0.25\right)}^4<{\left(0.35\right)}^3$