Find all real numbers $a$ that satisfy \[\frac{1}{a^3 + 7} -7 = \frac{-a^3}{a^3 + 7}.\]
Multiplying through by a^3 + 7 we have
1 - 7 (a^3 +7) = -a^3
1 - 7a^3 - 49 = -a^3
-7a^3 + a^3 = 49 - 1
-6a^3 = 48
a^3 = -48/6
a^3 = -8
a = -2
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