(0.25) ^ (x + 1) = 16
To solve this exponential equation, we need to isolate the variable (x) and solve for its value.
First, we can rewrite the equation using the definition of exponentiation:
0.25^(x + 1) = 16
Since 0.25 is equal to 1/4, we can substitute it:
(1/4)^(x + 1) = 16
Now, let's express both sides of the equation using the same base. Since 16 can be written as 2^4, we can rewrite the equation as:
(1/4)^(x + 1) = (2^4)
To simplify the equation further, we can rewrite 1/4 as 2^(-2):
(2^(-2))^(x + 1) = (2^4)
Using the power rule for exponential expressions, we can multiply the exponents:
2^(-2 * (x + 1)) = 2^4
Now, we have two exponential expressions with the same base. Therefore, the exponents must be equal:
-2 * (x + 1) = 4
Distributing the -2:
-2x - 2 = 4
Adding 2 to both sides:
-2x = 6
Dividing by -2:
x = -3
Therefore, the value of x that makes the equation true is -3.