# Star and Pound Operations

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

## Solution

In order to make it feasible for teachers to use these problems in their classwork, no solutions are publicly visible, so students cannot simply look up the answers. If you would like to view the solutions to these problems, you must have a Virtual Classroom subscription.## Similar Problems

### 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 = ?

### 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.

### 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.

### 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).

### Fractional Operation

### Hashtag Operation

### 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?

### Triangular Operation

The arrangement### Star Operation with Cubes

^{3}+ b

^{3}

^{2}- ab + b

^{2}