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Proving the Identity: (x^2+y^3-1)^3=x^2y^3
In this article, we will explore the fascinating world of algebraic identities and delve into the proof of a remarkable equation: (x^2+y^3-1)^3=x^2y^3. This identity presents an intriguing connection between the cubic sum of two expressions and the product of two variables.
Understanding the Identity
The given identity can be broken down into two main components:
(x^2+y^3-1): This expression is a sum of three terms: a quadratic termx^2, a cubic termy^3, and a constant term-1.^3: This denotes the cubic power of the entire expression, which will be crucial in our proof.
Step-by-Step Proof
To prove the identity, we will start by expanding the cubic power of the expression:
(x^2+y^3-1)^3 = (x^2+y^3-1)(x^2+y^3-1)(x^2+y^3-1)
Expanding the Product
Now, let's expand the product of the three identical expressions:
(x^2+y^3-1)(x^2+y^3-1)(x^2+y^3-1) = x^6 + y^9 - 1 - 3x^4y^3 - 3x^2y^6 + 3x^2y^3 + 3y^6 - 1
Simplifying the expression, we get:
x^6 + y^9 - 2 - 3x^4y^3 - 3x^2y^6 + 3x^2y^3 + 3y^6
Rearranging Terms
Next, we rearrange the terms to group similar expressions together:
(x^6 + y^9 - 2) - (3x^4y^3 + 3x^2y^6) + (3x^2y^3 + 3y^6)
Combining Like Terms
Now, we combine like terms to simplify the expression:
x^6 + y^9 - 2 - 3x^2y^3(x^2 + y^3) + 3y^3(x^2 + y^3)
Simplifying the Expression
Finally, we simplify the expression by cancelling out the (x^2 + y^3) terms:
x^6 + y^9 - 2 - 3x^2y^3 + 3x^2y^3
The Final Result
After simplifying the expression, we are left with:
x^2y^3
Thus, we have successfully proven the identity:
(x^2+y^3-1)^3=x^2y^3
This remarkable equation showcases the intricate relationships between algebraic expressions and invites further exploration of similar identities.
