Answer:
Therefore, the solution to the equation (2/3)^-2m x (3/2)^4 = (3/2)^-16 is m = -10.
Explanation:
To solve the equation (2/3)^-2m x (3/2)^4 = (3/2)^-16, we can use the properties of exponents and algebraic manipulation.
First, let's simplify the left side of the equation:
(2/3)^-2m x (3/2)^4
To simplify (2/3)^-2m, we can apply the property that states (a/b)^-n = (b/a)^n:
(2/3)^-2m = (3/2)^2m
Now we have:
(3/2)^2m x (3/2)^4
Next, we can apply the property that states (a^n) x (a^m) = a^(n+m):
(3/2)^2m x (3/2)^4 = (3/2)^(2m+4)
Now we can compare the left side of the equation to the right side:
(3/2)^(2m+4) = (3/2)^-16
Since the bases are the same, we can equate the exponents:
2m + 4 = -16
Now we can solve for m:
2m = -16 - 4
2m = -20
m = -10