To make the equation 5^m ⋅ (5−7)^m = 5^12 true, we need to simplify both sides of the equation and find the value of m that satisfies the equation.
Let's simplify the equation step by step:
5^m ⋅ (5−7)^m = 5^12
Since (5-7) = -2, we have:
5^m ⋅ (-2)^m = 5^12
Next, we can rewrite (-2)^m as (2^m) * (-1)^m:
5^m ⋅ (2^m) * (-1)^m = 5^12
Now, we can cancel out the common factors of 5 and 2 from both sides:
(-1)^m = (5^12) / (5^m * 2^m)
Simplifying further, we have:
(-1)^m = 5^(12-m) / 2^m
To make this equation true, we need to find a value of m that satisfies the equation. Since (-1)^m is either 1 or -1 depending on whether m is even or odd, we can write two separate equations:
If m is even: 1 = 5^(12-m) / 2^m
If m is odd: -1 = 5^(12-m) / 2^m
Now, we can solve these equations separately to find the value of m that makes the original equation true.