Show that
Question:

For $a l l a, b \in N$, we define $a * b=a^{3}+b^{3}$.

Show that $*$ is commutative but not associative.

Solution:

let $a=1, b=2 \in N$

$a^{*} b=1^{3}+2^{3}=9$

And $b * a=2^{3}+1^{3}=9$

Hence * is commutative.

Let c = 3

$\left(a^{*} b\right)^{*} c=9^{*} c=9^{3}+3^{3}$

$a^{*}\left(b^{*} c\right)=a^{*}\left(2^{3}+3^{3}\right)=1 * 35=1^{3}+35^{3}$

$\left(a^{*} b\right)^{*} c \neq a^{*}\left(b^{*} c\right)$

Hence * is not associative.