Final answer:
The equation 25^(x+3) = 1/5 can be solved by recognizing that 25 is 5^2 and rewriting the expression as 5^(2(x+3)) = 5^(-1), then equating exponents and solving for x to get x = -7/2.
Step-by-step explanation:
To solve the equation 25^(x+3) = 1/5, we must recognize that 25 is a power of 5. Specifically, 25 is 5 squared or 5^2. Subsequently, we can rewrite the equation as (5^2)^(x+3) = 5^(-1).
Using the laws of exponents, when we raise a power to a power, we multiply the exponents. So the left-hand side of the equation becomes 5^(2(x+3)). As we have the same base on both sides of the equation (5), we can equate the exponents:
2(x + 3) = -1
Now, solve for x:
- 2x + 6 = -1
- 2x = -1 - 6
- 2x = -7
- x = -7/2
Therefore, the solution to the equation 25^(x+3) = 1/5 is x = -7/2.