If \({\rm{A}} = {\rm{}}\frac{{0.216{\rm{\;}} + {\rm{\;}}0.008}}{{0.36{\rm{\;}} + {\rm{\;}}0.04{\rm{\;}} - {\rm{\;}}0.12}}\) and \({\rm{B}} = {\rm{}}\frac{{0.729{\rm{\;}} - {\rm{\;}}0.027}}{{0.81{\rm{\;}} + {\rm{\;}}0.09{\rm{\;}} + {\rm{\;}}0.27}}\) , then what is the value of (A 2+ B 2) 2?
1
Recognize cube/square patterns. For A, numerator is \((0.6)^3+(0.2)^3\) and denominator is \((0.6)^2-(0.6)(0.2)+(0.2)^2\). Using \(a^3+b^3=(a+b)(a^2-ab+b^2)\):
\[A=0.6+0.2=0.8\]
Similarly, B uses \(a^3-b^3=(a-b)(a^2+ab+b^2)\) with \(a=0.9,b=0.3\):
\[B=0.9-0.3=0.6\]
Then \(A^2+B^2=0.64+0.36=1\), so \((A^2+B^2)^2=\mathbf{1}\).
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