Question: User: 25^x \times 5^{x+1} = \frac{1}{5^2}
1. **State the problem:** Solve the equation $$25^x \times 5^{x+1} = \frac{1}{5^2}$$ for $x$.
2. **Rewrite the bases:** Note that $25 = 5^2$, so rewrite $25^x$ as $\left(5^2\right)^x = 5^{2x}$.
3. **Substitute:** The equation becomes $$5^{2x} \times 5^{x+1} = 5^{-2}$$ because $\frac{1}{5^2} = 5^{-2}$.
4. **Use exponent addition rule:** When multiplying with the same base, add exponents:
$$5^{2x} \times 5^{x+1} = 5^{2x + x + 1} = 5^{3x + 1}$$
5. **Set exponents equal:** Since the bases are equal and nonzero, set exponents equal:
$$3x + 1 = -2$$
6. **Solve for $x$:**
$$3x = -2 - 1$$
$$3x = -3$$
$$x = \frac{-3}{3}$$
$$x = -1$$
**Final answer:** $$x = -1$$