Without expanding the determinant, prove that `|(a,a^2,bc),(b,b^2,ca),(c,c^2,ab)|=|(1,a^2,a^3),(1,b^2,b^3),(1,c^2,c^3)|`

Explanation

To prove that the determinants \( |(a, a^2, bc), (b, b^2, ca), (c, c^2, ab)| \) and \( |(1, a^2, a^3), (1, b^2, b^3), (1, c^2, c^3)| \) are equal without expanding them, we can follow these steps:

### Step 1: Write the Left-Hand Side (LHS) Determinant
The LHS determinant is:
\[
D_1 = \begin{vmatrix}
a & a^2 & bc \\
b & b^2 & ca \\
...