%5E3-formula-example.jpeg "(a+b+c)^3 Formula Example")
(a + b + c)^3 Formula and Example
The formula for expanding the cube of a sum of three terms, (a + b + c)^3, is a fundamental concept in algebra. It is used in various mathematical problems, including those involving arithmetic, geometry, and calculus.
The Formula
The formula for expanding (a + b + c)^3 is:
(a + b + c)^3 = a^3 + b^3 + c^3 + 3a^2(b + c) + 3b^2(a + c) + 3c^2(a + b) + 6abc
This formula can be derived by using the distributive property of multiplication over addition and the binomial theorem.
Example
Let's consider an example to illustrate the use of this formula:
Find the value of (2 + 3 + 4)^3
Using the formula, we get:
(2 + 3 + 4)^3 = 2^3 + 3^3 + 4^3 + 3(2^2)(3 + 4) + 3(3^2)(2 + 4) + 3(4^2)(2 + 3) + 6(2)(3)(4)
= 8 + 27 + 64 + 3(4)(7) + 3(9)(6) + 3(16)(5) + 144
= 8 + 27 + 64 + 84 + 162 + 240 + 144
= 729
Therefore, the value of (2 + 3 + 4)^3 is 729.
Applications
The formula for expanding (a + b + c)^3 has various applications in mathematics and other fields, including:
- Algebraic manipulations: It is used to simplify complex algebraic expressions involving sums of cubes.
- Geometry: It is used to calculate the volumes of rectangular prisms and pyramids.
- Calculus: It is used to evaluate integrals and derivatives of functions involving sums of cubes.
In conclusion, the formula for expanding (a + b + c)^3 is a powerful tool in mathematics that has numerous applications in various fields.
