Proving the Identity: How to Show (a^3 b^3 c^3 - 3abc abc(a^2 b^2 c^2 - ab - bc - ac))

Mathematics is a field rich with identities and proofs. One such intriguing identity is:

[a^3 b^3 c^3 - 3abc abc(a^2 b^2 c^2 - ab - bc - ac)]

In this article, we will delve into the algebraic manipulation required to prove this identity. This involves expanding and simplifying the right-hand side (RHS) of the equation and demonstrating that it simplifies to the left-hand side (LHS).

Step 1: Expand the Right-Hand Side (RHS)

The RHS of the given identity is:

(abc(a^2 b^2 c^2 - ab - bc - ac))

Let's expand this expression step by step.

Expanding (a^2 b^2 c^2)

First, we distribute abc across each term inside the parentheses:

(abc cdot a^2 b^2 c^2 a^3 b^3 c^3) (abc cdot (-ab) -a^2 b^2 c) (abc cdot (-bc) -a b^2 c^2) (abc cdot (-ac) -a^2 b c^2)

Combining these results, we get:

(a^3 b^3 c^3 - a^2 b^2 c - a b^2 c^2 - a^2 b c^2)

Step 2: Simplify the Expression

Now, let's look at the entire RHS:

(a^3 b^3 c^3 - (a^2 b^2 c a b^2 c^2 a^2 b c^2))

Notice that we have several terms with similar variables but with different combinations of exponents. Let's group and simplify these terms:

(a^3 b^3 c^3) (-(a^2 b^2 c a b^2 c^2 a^2 b c^2) -a^2 b^2 c - a b^2 c^2 - a^2 b c^2)

Combining everything, we have:

(a^3 b^3 c^3 - 3abc)

Step 3: Final Rearrangement

We want to show that:

[a^3 b^3 c^3 - 3abc abc(a^2 b^2 c^2 - ab - bc - ac)]

To do this, we need to rearrange the terms on the RHS to match the LHS. Notice that the RHS simplifies to:

(a^3 b^3 c^3 - 3abc)

And the LHS is:

(a^3 b^3 c^3 - 3abc)

Therefore, both sides are equal, and we have proven the identity:

[a^3 b^3 c^3 - 3abc abc(a^2 b^2 c^2 - ab - bc - ac)]

Conclusion

We have shown through systematic algebraic manipulation that the identity holds true:

[a^3 b^3 c^3 - 3abc abc(a^2 b^2 c^2 - ab - bc - ac)]

This method of proving mathematical identities is a fundamental skill in algebra and is useful in various areas of mathematics and its applications.