1. ## Binary operations?

hey guy could somebody help me to solve this :
is the binary operation * defined for real numbers x,y by x*y=x^2y^2
is it commutative? is it associative?

thank you

2. Hello, boomshine57th!

Is the binary operation $\displaystyle *$ defined for real numbers $\displaystyle x,y$ by: .$\displaystyle x*y \:=\:x^2y^2$ ?

If $\displaystyle x,y$ are any real numbers, then: .$\displaystyle x*y \:=\:x^2y^2$ is a real number.

The operation is defined for real numbers.

Is it commutative?

If the operation is commutative, then: .$\displaystyle x*y \:=\:y*x$

We have: .$\displaystyle \begin{array}{cccc}x*y &=& x^2y^2 \\ y*x &=&y^2x^2 \end{array}$

Since $\displaystyle x^2y^2 = y^2x^2$, the operation is commutative.

Is it associative?

If the operation is associative, then: .$\displaystyle x*(y*z) \:=\:(x*y)*z$

We have: .$\displaystyle \begin{array}{ccccccc} x*(y*z) &=& x*(y^2z^2) &=& x^2(y^2z^2)^2 &=& x^2y^4z^4 \\ (x*y)*z &=& (x^2y^2)*z &=& (x^2y^2)^2 z^2 &=& x^4y^4z^2\end{array}\quad\hdots$ not equal

The operation is not associative.

3. ## thank you

thank you soooo much! you are a life saver