Fractional OperationPro Problems > Math > Number and Quantity > Number Theory > Star Operations
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The operation * is defined as follows: x * y = (ax + b)(by + a), for some real numbers a and b. Under what conditions is this operation commutative?
There exists a star (*) operation for which the following equations are true:
2 * 3 = 10
5 * 2 = 35
3 * 6 = 27
8 * 7 = 120
Find a star operation rule which satisfies all of these equations, and then use your rule to calculate the following:
10 * 11 = ?
x * y = x2 - y2 + kx - hy. If 5 * 2 = 48 and 4 * 3 = 30, find the ordered pair (h, k).
If the operation ♦ is defined as follows: a ♦ b = a2 - b2 , and 18 ♦ (x ♦ 8) = 6 ♦ 1, find all possible values of x.
If a * b = (a + b)2, and a # b = (a - b)2, find x such that (2 * x) # (x * 1) = 81
x ♦ 2x = 2x ♦ x.
Note that this operation is not commutative; the order of operands cannot be exchanged without altering the result of the operation.