We are tasked with solving the equation $4^x = 16^{x+1}$ to find the value of $x$.

Step-by-Step Solution:

1. Rewrite the bases with the same powers if possible:

   • $4 = 2^2$ and $16 = 2^4$. So, rewrite the equation in terms of base $2$: $(2^2)^x = (2^4)^{x+1}.$

2. Simplify the exponents:

   • Using the rule $(a^m)^n = a^{m \cdot n}$, the equation becomes: $2^{2x} = 2^{4(x+1)}.$

3. Equate the exponents (since the bases are equal): $2x = 4(x + 1).$

4. Expand and simplify:

   • Expand $4(x + 1)$: $2x = 4x + 4.$
   • Rearrange the terms to isolate $x$: $2x - 4x = 4,$ $-2x = 4.$

5. Solve for $x$: $x = \frac{-4}{2} = -2.$

Final Answer:

$x = -2$

Would you like further clarification or a breakdown of any specific step?

Related Questions:

1. How do you simplify expressions involving exponential equations with different bases?
2. What other techniques can be used to solve equations like $a^{f(x)} = b^{g(x)}$?
3. How does the rule $(a^m)^n = a^{m \cdot n}$ apply to solving exponential equations?
4. Why is it important to rewrite the bases of exponents in terms of a common base?
5. What happens if you have logarithms in such equations?

Tip:

Always rewrite the bases in exponential equations to the same base if possible; it simplifies the problem significantly!