If $$a^2+1=a$$, then the value of $$a^3$$ is
$$a^2+1=a$$
$$a^2-a=-1$$ and $$a^2=a-1$$
Therefore
$$a^3=a^2.a$$
     =$$\left(a-1\right)a$$
     = $$a^2-a$$
      = -1
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