If 16(4)^2d = (1\2)^d. Solve for d
To solve for d, we can start by simplifying both sides of the equation.
16(4)^2d = (1/2)^d
First, simplify the left side:
16(4)^2d = 16 * 16^d = 16^(1 + 2d)
Next, simplify the right side:
(1/2)^d = 2^(-d) = (2^-1)^d = 2^(-d)
Now, the equation becomes:
16^(1 + 2d) = 2^(-d)
Since both 16 and 2 can be written as powers of 2, we can rewrite the equation:
(2^4)^(1 + 2d) = 2^(-d)
Using the power rule of exponents (a^(bc) = (a^b)^c):
2^(4(1 + 2d)) = 2^(-d)
Simplify both sides:
2^(4 + 8d) = 2^(-d)
Now that the bases are equal, the exponents must be equal:
4 + 8d = -d
Combine like terms:
8d + d = -4
9d = -4
Now, solve for d:
d = -4/9
Therefore, the solution is d = -4/9.