Answer

Correct Option: (iii) Equivalent in power

Explanation

Multi-tape TM vs Single-tape TM:

• Power: Equivalent. Any $k$-tape TM can be simulated by single-tape TM with at most quadratic slowdown (or $O(t^2)$). Both accept exactly RE languages; deciders both accept Recursive languages.
• Efficiency: Multi-tape can be faster (e.g., palindrome $O(n)$ vs $O(n^2)$ on single tape), but not more powerful in computability.
• Construction: Single tape encodes $k$ tapes via tracks ($k$-track tape) + markers for heads.

Therefore they are equally powerful (Turing-equivalent).

Final Answer

Equivalent in power → Option (iii).