%5E3-formula-proof.jpeg "(a+b)^3 Formula Proof")
The Proof of (a + b)³
The formula (a + b)³ is a fundamental concept in algebra, used to expand the cube of a binomial expression. Here's a breakdown of how to prove this formula:
Understanding the Formula
The formula states that:
(a + b)³ = a³ + 3a²b + 3ab² + b³
This means that when you cube the sum of two terms (a + b), you get the sum of four terms: the cube of the first term, three times the square of the first term multiplied by the second term, three times the first term multiplied by the square of the second term, and the cube of the second term.
Proof using Expansion and Distribution
We can prove this formula by expanding the expression and using the distributive property:
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Expand the expression: (a + b)³ = (a + b)(a + b)(a + b)
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Multiply the first two factors: (a + b)(a + b) = a² + ab + ba + b² = a² + 2ab + b²
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Multiply the result by the third factor: (a² + 2ab + b²)(a + b) = a³ + 2a²b + ab² + a²b + 2ab² + b³
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Combine like terms: a³ + 2a²b + ab² + a²b + 2ab² + b³ = a³ + 3a²b + 3ab² + b³
Therefore, the formula (a + b)³ = a³ + 3a²b + 3ab² + b³ is proven.
Visualizing the Proof
You can also visualize the proof using a cube. Imagine a cube with side length (a + b). You can divide this cube into smaller cubes with side lengths a and b. This visual representation helps to see how the formula breaks down into the individual terms.
Applications of the Formula
The (a + b)³ formula has wide applications in various areas, including:
- Algebraic simplification: This formula helps simplify complex expressions involving cubes of binomials.
- Calculus: It's used in finding derivatives and integrals of functions involving binomial cubes.
- Geometry: This formula can be applied to calculate the volume of certain geometric shapes.
- Physics and engineering: It's used in solving problems related to motion, force, and energy.
Understanding the proof of this formula is crucial for mastering algebraic manipulations and applying it to solve various problems in different fields.