# Fractional Operation

Pro Problems > Math > Number and Quantity > Number Theory > Star Operations## Fractional Operation

## Solution

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### Commutative Operation

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?

### Guess the Star

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 = ?

### h and k

x * y = x^{2} - y^{2} + kx - hy. If 5 * 2 = 48 and 4 * 3 = 30, find the ordered pair (h, k).

### Diamond Operation

If the operation ♦ is defined as follows: a ♦ b = a^{2} - b^{2} , and 18 ♦ (x ♦ 8) = 6 ♦ 1, find all possible values of x.

### Star and Pound Operations

If a * b = (a + b)^{2}, and a # b = (a - b)^{2}, find x such that (2 * x) # (x * 1) = 81

### Hashtag Operation

### Star Operation with Cubes

^{3}+ b

^{3}

^{2}- ab + b

^{2}

### Non-Commutative Operation

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.