Understanding the Expansion of (a b)^3: A Comprehensive Guide

Introduction

The expression a^3 3a^2b 3ab^2 b^3 represents the expanded form of (a b)^3. This article will explore the process of expanding the binomial expression and the significance of the terms involved. We will use the binomial theorem and the distributive property to demonstrate how to simplify and expand such expressions.

Using the Binomial Theorem

The binomial theorem is a powerful mathematical tool used to expand binomials to any power. For the expression (a b)^3, the theorem provides a straightforward method for expansion. The binomial theorem states that:

(a b)^n Σ [C(n, k) * a^(n-k) * b^k], where C(n, k) is the binomial coefficient.

Applying this to (a b)^3:

(a b)^3 C(3, 0) * a^3 C(3, 1) * a^2 * b C(3, 2) * a * b^2 C(3, 3) * b^3

Calculating the binomial coefficients:

C(3, 0) 1 C(3, 1) 3 C(3, 2) 3 C(3, 3) 1

Thus, the expansion simplifies to:

(a b)^3 a^3 3a^2b 3ab^2 b^3

Using the Distributive Property

Another method to expand (a b)^3 is by using the distributive property. This involves multiplying the binomial (a b) by itself three times:

(a b)^3 (a b) * (a b) * (a b)

Performing the multiplication step-by-step:

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

Combining like terms:

a^3 3a^2b 3ab^2 b^3

Conclusion

Both the binomial theorem and the distributive property yield the same result, demonstrating that the expansion of (a b)^3 is given by the formula:

a^3 3a^2b 3ab^2 b^3

This expression is a fundamental concept in algebra with applications in various fields such as physics, engineering, and computer science. Understanding how to expand binomials using these methods will greatly enhance your problem-solving skills in mathematics.

Key Takeaways

The binomial theorem and distributive property can be used to expand expressions of the form (a b)^3. The expanded form is a^3 3a^2b 3ab^2 b^3. The binomial coefficients in the binomial theorem provide a systematic way to expand the expression.