%5E5-math-formula.jpeg "(a+b)^5 Math Formula")
The Power of Binomials: Understanding (a+b)^5
In algebra, binomials are algebraic expressions consisting of two terms, such as a + b. Raising a binomial to a power, like (a+b)^5, can be a daunting task, but fear not! In this article, we'll explore the formula and step-by-step process to expand this expression.
The Formula:
The formula to expand (a+b)^n, where n is a positive integer, is known as the Binomial Theorem. For our case, n = 5, so the formula is:
(a+b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5
This formula might look intimidating, but let's break it down step by step.
Step-by-Step Expansion:
To expand (a+b)^5, we'll use the principle of repeated multiplication. We'll multiply the binomial by itself five times, using the distributive property of multiplication over addition.
(a+b)^5 = (a+b) × (a+b) × (a+b) × (a+b) × (a+b)
Multiplication 1: (a+b) × (a+b) = a^2 + 2ab + b^2
Multiplication 2: (a^2 + 2ab + b^2) × (a+b) = a^3 + 3a^2b + 3ab^2 + b^3
Multiplication 3: (a^3 + 3a^2b + 3ab^2 + b^3) × (a+b) = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4
Multiplication 4: (a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4) × (a+b) = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5
And there you have it! The expanded form of (a+b)^5.
Conclusion:
Expanding (a+b)^5 might seem like a complex task, but by using the Binomial Theorem and breaking it down step by step, we can easily arrive at the final expression. This formula is essential in various mathematical fields, including algebra, combinatorics, and calculus. Now, go forth and conquer more complex binomial expansions!
