Prove: \[ (x^{a-b})^{a+b}(x^{b-c})^{b+c}(x^{c-a})^{c+a} = 1 \]
Proof
\[ = x^{(a-b)(a+b)} \cdot x^{(b-c)(b+c)} \cdot x^{(c-a)(c+a)} \]
\[ = x^{(a^2-b^2) + (b^2-c^2) + (c^2-a^2)} \]
\[ = x^0 \]
\[ = 1 \]
\[ = x^{(a-b)(a+b)} \cdot x^{(b-c)(b+c)} \cdot x^{(c-a)(c+a)} \]
\[ = x^{(a^2-b^2) + (b^2-c^2) + (c^2-a^2)} \]
\[ = x^0 \]
\[ = 1 \]