Maker-Breaker Rado games for equations with radicals

We study two-player positional games where Maker and Breaker take turns to select a previously unoccupied number in $\{1,2,\dots ,n\}$. Maker wins if the numbers selected by Maker contain a solution to the equation $$x_1^{1\mathrm{/}\mathrm{\ell }}+\cdots +x_k^{1\mathrm{/}\mathrm{\ell }}=y^{1\mathrm{/}\mathrm{\ell }}$$ where $k$ and $\mathrm{\ell }$ are integers with $k\ge 2$ and $\mathrm{\ell }\mathrm{\ne }0$, and Breaker wins if they can stop Maker. Let $f(k,\mathrm{\ell })$ be the smallest positive integer $n$ such that Maker has a winning strategy when $x_1,\dots ,x_k$ are not necessarily distinct, and let $f^{\ast }(k,\mathrm{\ell })$ be the smallest positive integer $n$ such that Maker has a winning strategy when $x_1,\dots ,x_k$ are distinct. When $\mathrm{\ell }\ge 1$, we prove that, for all $k\ge 2$, $f(k,\mathrm{\ell })=(k+2{)}^{\mathrm{\ell }}$ and $f^{\ast }(k,\mathrm{\ell })=(k^2+3{)}^{\mathrm{\ell }}$; when $\mathrm{\ell }\le -1$, we prove that $f(k,\mathrm{\ell })=[k+{\mathrm{\Theta }}_k(1){]}^{-\mathrm{\ell }}$ and $f^{\ast }(k,\mathrm{\ell })=[\exp (O_k(k\log k)){]}^{-\mathrm{\ell }}$. Our proofs use elementary combinatorial arguments as well as results from number theory and arithmetic Ramsey theory.