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(a+b)^3 Expansion Formula: A Comprehensive Guide
The (a+b)^3 expansion formula is a fundamental concept in algebra and is widely used in various mathematical calculations. In this article, we will delve into the details of this formula, its derivation, and some examples to illustrate its application.
What is the (a+b)^3 Expansion Formula?
The (a+b)^3 expansion formula is a mathematical expression that represents the cube of the sum of two variables, a and b. It is defined as:
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
This formula is a crucial tool for simplifying complex algebraic expressions and is used extensively in various mathematical disciplines, including algebra, geometry, and calculus.
Derivation of the (a+b)^3 Expansion Formula
The (a+b)^3 expansion formula can be derived using the binomial theorem, which states that:
(a+b)^n = a^n + na^(n-1)b + ... + nab^(n-1) + b^n
where n is a positive integer.
To derive the (a+b)^3 expansion formula, we can substitute n=3 into the binomial theorem:
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
This formula can be verified by expanding the cube of the sum of a and b using the distributive property of multiplication over addition.
Examples of the (a+b)^3 Expansion Formula
Example 1:
Find the value of (2+3)^3 using the (a+b)^3 expansion formula.
Solution:
Substitute a=2 and b=3 into the formula:
(2+3)^3 = 2^3 + 3(2^2)(3) + 3(2)(3^2) + 3^3 (2+3)^3 = 8 + 36 + 54 + 27 (2+3)^3 = 125
Example 2:
Simplify the expression (x+2)^3 using the (a+b)^3 expansion formula.
Solution:
Substitute a=x and b=2 into the formula:
(x+2)^3 = x^3 + 3x^2(2) + 3x(2^2) + 2^3 (x+2)^3 = x^3 + 6x^2 + 12x + 8
Conclusion
In conclusion, the (a+b)^3 expansion formula is a powerful tool for simplifying complex algebraic expressions and is a fundamental concept in algebra. Its application is diverse, ranging from basic algebra to advanced mathematical disciplines. By mastering this formula, you will be able to tackle complex mathematical problems with ease and confidence.
