Answer; To find the possible values of m for which 3*m = 7, we need to substitute the given equation into the definition of the binary operation *.
The definition of the binary operation * is x * y = x^2 - y^2y.
So, substituting 3 for x and m for y, we have:
3 * m = 3^2 - m^2m
Simplifying this equation, we get:
9 - m^2m = 7
Rearranging the equation, we have:
-m^2m = 7 - 9
-m^2m = -2
Dividing both sides of the equation by -1, we get:
m^2m = 2
Taking the square root of both sides, we have:
m * sqrt(m) = sqrt(2)
Now, we need to solve for m.
Since the square root of a number can be positive or negative, we consider both cases:
Case 1: m * sqrt(m) = sqrt(2)
By squaring both sides of the equation, we have:
m^2 * m = 2
Simplifying further, we get:
m^3 = 2
Taking the cube root of both sides, we have:
m = ∛2
Case 2: m * sqrt(m) = -sqrt(2)
By squaring both sides of the equation, we have:
m^2 * m = -2
Simplifying further, we get:
m^3 = -2
Taking the cube root of both sides, we have:
m = -∛2
Therefore, the possible values of m for which 3 * m = 7 are:
m = ∛2 or m = -∛2.