Which number is bigger: $\sqrt[10]{2}$ or $1.2$? What is the general method for finding such inequalities? I have some more problems of this kind in the text I am using.
 A: By the AM-GM Inequality,
$$\frac{2+1+1+1+1+1+1+1+1+1}{10} \geq \sqrt[10] 2$$
$$\frac{11}{10} \geq \sqrt[10] 2 \Rightarrow 1.2 \geq \sqrt[10] 2$$
The AM-GM inequality sometimes works when you have an n-th root but it depends on the problem.
A: There is no general method. There have been numerous books written on just inequalities. For example, see Inequalities: Theorems, Techniques and Selected Problems. However, oftentimes, simple inequalities like yours may be approached by applying some subtle manipulations, much as mathlove's excellent answer has done. Consider the following example.
Example: Compare $6^{1/4}$ and $4^{1/3}$ without using a calculator. Is one greater than the other?
Answer. Observe that
$$
6^{\frac{1}{4}}=6^{\frac{3}{12}}=(6^3)^{\frac{1}{12}}=216^{\frac{1}{12}}
$$
and
$$
4^{\frac{1}{3}}=4^{\frac{4}{12}}=(4^4)^{\frac{1}{12}}=256^{\frac{1}{12}}.
$$
Since $256$ and $216$ are both being raised to the same power, and $256>216$, we can clearly see that $4^{1/3} > 6^{1/4}$ since
$$
4^{\frac{1}{3}} = 256^{\frac{1}{12}} > 216^{\frac{1}{12}} = 6^{\frac{1}{4}}.
$$

Knowing that it helps to make such manipulations comes with time and practice. Notice that it would otherwise be quite difficult to compare the values in the example as $4^{1/3}\approx 1.5874$ and $6^{1/4}\approx 1.5651$. 
A: Using the binomial theorem, 
$$1.2^{10}=\left(1+\frac 15\right)^{10}\gt 1+\binom{10}{1}\frac 15=3\gt 2\Rightarrow 1.2\gt \sqrt[10]2.$$
A: Try raising $1.2$ to higher and higher powers:
$$
(1.2)^2=1.44 > 1.4,
$$
$$
(1.2)^4 > (1.4)^2 = 1.96 > 1.9,
$$
$$
(1.2)^5 > (1.2)(1.9)=2.28 > 2.
$$
So already $(1.2)^5 > 2$, hence $1.2 > \sqrt[5]{2} > \sqrt[10]{2}$.
A: $\sqrt[10]2\lt\sqrt[8]2$. $\sqrt[8]2$ is easy to compute by $\sqrt{\sqrt{\sqrt2}}$, which is certainly $\lt 1.2$.
A: Think of an investment that returns $20\%$ per year (and let me know if you can find one...).  Simple interest would return $100\%$ -- i.e. double -- in five years, hence quadruple in ten.  Compound interest, which is what $1.2^{10}$ represents, necessarily does better, so $1.2$ is quite a bit larger than $\sqrt[10]2$.
