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The Cartan formula $$Sq^i(xy)=\Sigma _{j+k=i}Sq^j(x)Sq^k(y)$$ together with the instability condition $$Sq^d(\alpha)=\alpha ^2 \mbox{ if $d=deg(\alpha )$}, Sq^i(\alpha )=0 \mbox{ if $d>deg(\alpha )$}$$ and the property $Sq^0=id$ (see, e.g. https://en.wikipedia.org/wiki/Steenrod_algebra) give $Sq^i (\alpha ^j)= \binom{j}{i}\alpha ^{i+j} $.