1. The problem is to simplify the expression $5^{3125^{3125}}$. 2. This is an exponential expression where the exponent itself is a power: $5^{(3125^{3125})}$. 3. First, recognize that $3125$ can be expressed as a power of 5: $3125 = 5^5$. 4. Substitute this back into the exponent: $$3125^{3125} = (5^5)^{3125}$$ 5. Use the power of a power rule: $\left(a^m\right)^n = a^{m \times n}$. 6. So, $$ (5^5)^{3125} = 5^{5 \times 3125} = 5^{15625} $$ 7. Now the original expression becomes: $$ 5^{3125^{3125}} = 5^{5^{15625}} $$ 8. This is the simplified form, expressing the original number as a power of 5 with a very large exponent. Final answer: $$ 5^{5^{15625}} $$