Which prime factorization-based justification correctly supports the simplification: $\sqrt{200}=10\sqrt{2}$?

Factors: $2\cdot 2\cdot 2\cdot 5\cdot 5$. Extract pairs ($2$ and $5$) for $10$ outside; leave the unpaired $2$ inside.

Factors: $2\cdot 10\cdot 10$. Extract the pair of $10$s outside; leave the $2$ inside.

Factors: $2\cdot 2\cdot 5\cdot 5$. Multiply to get $100$ ($10$ outside), then add a $2$ inside to balance.

Factors: $2\cdot 2\cdot 2\cdot 5\cdot 5$. Divide the prime factors by two to get $10$ outside and $2$ inside.