-4b%5E(2))%5E(3)%2B64(b%5E(2)-4c%5E(2))%5E(3)%2B(16c%5E(2)-a%5E(2))%5E(3))%2F((a-2b)%5E(3)%2B(2b-4c)%5E(3)%2B(4c-a)%5E(3)).jpeg "((a^(2)-4b^(2))^(3)+64(b^(2)-4c^(2))^(3)+(16c^(2)-a^(2))^(3))/((a-2b)^(3)+(2b-4c)^(3)+(4c-a)^(3))")
Exploring a Complex Algebraic Expression
This article delves into the intriguing algebraic expression:
(a^(2)-4b^(2))^(3)+64(b^(2)-4c^(2))^(3)+(16c^(2)-a^(2))^(3))/((a-2b)^(3)+(2b-4c)^(3)+(4c-a)^(3))
Let's break down this complex expression and uncover its hidden beauty.
Understanding the Structure
The expression consists of two main parts:
- Numerator: (a^(2)-4b^(2))^(3)+64(b^(2)-4c^(2))^(3)+(16c^(2)-a^(2))^(3)
- Denominator: (a-2b)^(3)+(2b-4c)^(3)+(4c-a)^(3)
Both the numerator and denominator are cubic expressions, with each term representing the cube of a specific binomial.
Simplifying the Expression
To simplify this expression, we need to exploit the properties of cubic equations and factor out common terms. Let's focus on the numerator first:
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Factoring by Grouping: We can group the first and third terms, and the second term separately. Notice that the first and third terms share a common factor of (a^2 - 16c^2), and the second term can be factored as 64(b^2 - 4c^2)^3.
(a^(2)-4b^(2))^(3)+(16c^(2)-a^(2))^(3) + 64(b^(2)-4c^(2))^(3)
This can be rewritten as:
(a^2 - 16c^2)[(a^2 - 4b^2)^2 + (a^2 - 16c^2)(a^2 - 4b^2) + (16c^2 - a^2)^2] + 64(b^2 - 4c^2)^3
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Recognizing a pattern: The expression in the square brackets resembles a cubic identity:
x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz)
Let's substitute:
- x = a^2 - 4b^2
- y = a^2 - 16c^2
- z = 16c^2 - a^2
Notice that x + y + z = 0. Therefore, the expression in the square brackets equals -3xyz.
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Simplifying further:
(a^2 - 16c^2)[ -3(a^2 - 4b^2)(a^2 - 16c^2)(16c^2 - a^2)] + 64(b^2 - 4c^2)^3
(a^2 - 16c^2)(-3)(a^2 - 4b^2)(a^2 - 16c^2)(16c^2 - a^2) + 64(b^2 - 4c^2)^3
-3(a^2 - 4b^2)(a^2 - 16c^2)^2(16c^2 - a^2) + 64(b^2 - 4c^2)^3
Simplifying the Denominator
Let's apply a similar strategy to the denominator.
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Factoring the cubes: The denominator is a sum of cubes. We can use the following identity:
a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc)
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Substituting:
- a = a - 2b
- b = 2b - 4c
- c = 4c - a
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Simplifying: Notice that a + b + c = 0, and therefore the denominator simplifies to:
(a - 2b)^2 + (2b - 4c)^2 + (4c - a)^2 - (a - 2b)(2b - 4c) - (a - 2b)(4c - a) - (2b - 4c)(4c - a)
Putting it All Together
Now, we have simplified both the numerator and denominator:
- Numerator: -3(a^2 - 4b^2)(a^2 - 16c^2)^2(16c^2 - a^2) + 64(b^2 - 4c^2)^3
- Denominator: (a - 2b)^2 + (2b - 4c)^2 + (4c - a)^2 - (a - 2b)(2b - 4c) - (a - 2b)(4c - a) - (2b - 4c)(4c - a)
The resulting expression is still quite complex, but we've managed to simplify it significantly. We can further investigate this expression by analyzing its factors and potentially identifying common factors that could be canceled out.
Conclusion
This exploration has demonstrated how powerful algebraic manipulation can be in simplifying complex expressions. By applying factoring techniques, recognizing patterns, and utilizing identities, we have made significant progress in understanding the structure of this intricate algebraic expression. Further investigation is necessary to determine whether further simplification is possible.