What are the formulas sum of perfect cubes and difference of perfect cubes?
bobpursley said to use them in one of my other posts but I don't know those formulas.
The formulas for the sum of perfect cubes and the difference of perfect cubes are derived from algebraic expressions. Let's start by understanding what a perfect cube is.
A perfect cube is a number that is obtained by multiplying an integer by itself twice. For example, 2 x 2 x 2 = 8, so 8 is a perfect cube.
Formula for the sum of perfect cubes:
The sum of two perfect cubes, a³ + b³, can be factored into (a + b)(a² - ab + b²). This is known as the sum of cubes formula. It states that the sum of two perfect cubes is equal to the product of their cube roots, multiplied by a quadratic expression.
Formula for the difference of perfect cubes:
Similarly, the difference of two perfect cubes, a³ - b³, can be factored into (a - b)(a² + ab + b²). This is called the difference of cubes formula. It states that the difference between two perfect cubes is equal to the product of their cube roots, multiplied by a quadratic expression.
To use these formulas, you simply need to identify the values of a and b in the given expression and substitute them into the appropriate formula. Let's look at an example:
Example 1: Find the sum of 5³ + 2³.
Using the sum of cubes formula, we have:
a = 5 and b = 2, so the formula becomes (5 + 2)(5² - (5)(2) + 2²).
Simplifying this expression, we get (7)(25 - 10 + 4) = (7)(19) = 133. So, 5³ + 2³ = 133.
Example 2: Find the difference of 8³ - 2³.
Using the difference of cubes formula, we have:
a = 8 and b = 2, so the formula becomes (8 - 2)(8² + (8)(2) + 2²).
Simplifying this expression, we get (6)(64 + 16 + 4) = (6)(84) = 504. So, 8³ - 2³ = 504.
In summary, the sum of perfect cubes formula is (a + b)(a² - ab + b²), and the difference of perfect cubes formula is (a - b)(a² + ab + b²). To use these formulas, substitute the given values of a and b into their respective formulas and simplify the resulting expression to find the answer.