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If $b$ and $c$ are elements in a group $G$, and if $b^5=c^3=e$, where $e$ is the unit element of $G$, then the inverse of $b^2 c b^4 c^2$ must be

  1. $b^4 c^2 b^2 c$
  2. $c^2 b^4 c b^2$
  3. $c b^2 c^2 b^4$
  4. $c b c^2 b^3$
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All India Mock Test 2 - Solutions Part 3

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We just need to line up the powers of $b$ and $c$ in reverse order, using that $c^{-k}=c^{3-k}$ and $b^{-\ell}=b^{5-\ell}$.
$$
b^2 c b^4 c^2 \text { gives us } c b c^2 b^3 \text {. }
$$
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