Cycle notation: Inverse of $(1348)$ in $S_9$ ( Find the mistake)

Question

Find inverse of $(1348)$ in $S_9.$

Here, we are using the cycle notation.

Clearly, the inverse is $(8431).$ Since $(1348)\circ (8431)$ will have $$8\rightarrow 4 \rightarrow 8 $$ $$4\rightarrow 3 \rightarrow 4 $$ $$3\rightarrow 1 \rightarrow 3 $$ $$1\rightarrow 8 \rightarrow 1 $$

So $(1348)\circ (8431)=e$

But we also have $(1348)\circ(1348)\circ (1348)\circ (1348)=e.$

So the inverse of $(1348)$ is $(1348)^3=(1348)\circ(1348)\circ (1348)=(8134).$

$$1\rightarrow 3 \rightarrow 4 \rightarrow 8 $$ $$3\rightarrow 4 \rightarrow 8 \rightarrow 1 $$ $$4\rightarrow 8 \rightarrow 1 \rightarrow 3 $$ $$8\rightarrow 1 \rightarrow 3 \rightarrow 4$$

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Clearly $(8134)$ is not the inverse of $(1348).$ What mistake have I made?

Thanks in advance!

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EDIT: I think I got the mistake, so after $(1348)^3,$ we get $$1\rightarrow 8 $$ $$3\rightarrow 1 $$ $$4\rightarrow 3 $$ $$8\rightarrow 4 $$

So, the answer is $$(1\rightarrow 8\rightarrow 4\rightarrow 3)=(1843)=(8431)$$

Answer 1

Here $1 \mapsto 3 \mapsto 4 \mapsto 8$ is the result of applying the permutation $3$ times.

So this means in $(1348)^3$, $1$ is mapped to $8$. Reading the others, we get $3\mapsto 1, 4\mapsto 3, 8\mapsto 4.$

If we convert that to this notation, we get $(1348)^3 = (1843),$ which is the same as $(8413).$

When you wrote $(8134)$, I think you just read the right-most numbers, which is not the right way to interpret it.