(a+b)^3 Simplify

2 min read Jun 16, 2024

Simplifying (a + b)^3

The expression (a + b)^3 represents the cube of the binomial (a + b). This means multiplying the binomial by itself three times:

(a + b)^3 = (a + b) * (a + b) * (a + b)

To simplify this, we can use the distributive property and some algebraic manipulations. Here's how:

Step 1: Expand the first two binomials

(a + b) * (a + b) = a(a + b) + b(a + b)
= a^2 + ab + ba + b^2
= a^2 + 2ab + b^2 (Since ab and ba are the same)

Step 2: Multiply the result by the remaining (a + b)

(a^2 + 2ab + b^2) * (a + b) = a(a^2 + 2ab + b^2) + b(a^2 + 2ab + b^2)
= a^3 + 2a^2b + ab^2 + a^2b + 2ab^2 + b^3
= a^3 + 3a^2b + 3ab^2 + b^3

Final Result

Therefore, the simplified form of (a + b)^3 is:

** (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3**

Key points

The expansion of (a + b)^3 follows a pattern: the exponents of 'a' decrease from 3 to 0, while the exponents of 'b' increase from 0 to 3.
The coefficients of the terms are determined by the binomial theorem: The coefficients are 1, 3, 3, 1, which correspond to the fourth row of Pascal's Triangle.

This simplified form is crucial in various mathematical contexts, including algebra, calculus, and physics. Understanding the expansion of (a + b)^3 helps in solving equations, performing complex calculations, and comprehending abstract mathematical concepts.