Is there a natural number $n$ for which $\sqrt[n]{22-10\sqrt7}=1-\sqrt7$ Is there a natural number $n$ for which $$\sqrt[n]{22-10\sqrt7}=1-\sqrt7$$
My idea was to try to express $22-10\sqrt7$ as something to the power of $2$, but it didn't work $$22-10\sqrt7=22-2\times5\times\sqrt7$$ Since $5^2=25, \sqrt7^2=7$ and $25+7\ne22$. What else can we try?

Note that $2.5\lt\sqrt7\lt2.7$, so $-1.7\lt1-\sqrt7\lt-1.5$, while $-5\lt22-10\sqrt7\lt-3$. The only possible integer power of $n$ for which $(1-\sqrt7)^n=22-10\sqrt7$ is $n=3$. (The power cannot be even, since even powers are non-negative, and if $n\ge5$ then $(-1.5)^n\lt-5$.)
This doesn't prove that $n=3$ satisfies the equation, of course. It merely tells us what needs to be checked, namely
$$(1-\sqrt7)^3=1-3\sqrt7+3\cdot7-7\sqrt7=22-10\sqrt7$$
