Solve for $x$: $$5^{x + 4} = 125^x.$$
Writing the right side with $5$ as the base, we have $125^x = (5^3)^x = 5^{3x}$, so our equation is: $$5^{x + 4} = 5^{3x}.$$Then, setting the exponents equal, we obtain $$x + 4 = 3x.$$This yields $2x = 4 \implies \boxed{x = 2}$
2
Level 2
Algebra
Problem: Solve for $x$: $$5^{x + 4} = 125^x.$$
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the equation, first, recognize that 125 can be expressed as a power of 5. Specifically, \(125 = 5^3\). Therefore, we can rewrite the equation as:

$$5^{x + 4} = (5^3)^x.$$

Next, apply the power of a power property of exponents, which states that \((a^m)^n = a^{m \cdot n}\). This allows us to simplify the right side:

$$(5^3)^x = 5^{3x}.$$

Now, the equation becomes:

$$5^{x + 4} = 5^{3x}.$$

Since the bases are the same, we can set the exponents equal to each other:

$$x + 4 = 3x.$$

To isolate \(x\), subtract \(x\) from both sides:

$$4 = 3x - x,$$

which simplifies to:

$$4 = 2x.$$

Next, divide both sides by 2 to solve for \(x\):

$$x = \frac{4}{2} = 2.$$

Thus, the solution to the equation is \(x = 2\).<|END_OF_LATENT|> The final answer is $2$.