Use Product Rule : \({x}^{a}{x}^{b}={x}^{a+b}\).
\[(2+a)(2+{a}^{2}){(2-{a}^{2})}^{2}=23\]
Expand.
\[16-16{a}^{2}+4{a}^{4}+8{a}^{2}-8{a}^{4}+2{a}^{6}+8a-8{a}^{3}+2{a}^{5}+4{a}^{3}-4{a}^{5}+{a}^{7}=23\]
Simplify \(16-16{a}^{2}+4{a}^{4}+8{a}^{2}-8{a}^{4}+2{a}^{6}+8a-8{a}^{3}+2{a}^{5}+4{a}^{3}-4{a}^{5}+{a}^{7}\) to \(16-8{a}^{2}-4{a}^{4}+2{a}^{6}+8a-4{a}^{3}-2{a}^{5}+{a}^{7}\).
\[16-8{a}^{2}-4{a}^{4}+2{a}^{6}+8a-4{a}^{3}-2{a}^{5}+{a}^{7}=23\]
Move all terms to one side.
\[16-8{a}^{2}-4{a}^{4}+2{a}^{6}+8a-4{a}^{3}-2{a}^{5}+{a}^{7}-23=0\]
Simplify \(16-8{a}^{2}-4{a}^{4}+2{a}^{6}+8a-4{a}^{3}-2{a}^{5}+{a}^{7}-23\) to \(-7-8{a}^{2}-4{a}^{4}+2{a}^{6}+8a-4{a}^{3}-2{a}^{5}+{a}^{7}\).
\[-7-8{a}^{2}-4{a}^{4}+2{a}^{6}+8a-4{a}^{3}-2{a}^{5}+{a}^{7}=0\]
No root was found algebraically. However, the following root(s) were found by numerical methods.
\[a=1.759464\]
a=1.759464263916
