## Understanding the (a-b)³ Formula

The formula (a-b)³ is a common algebraic expression used to expand the cube of a binomial. It represents the product of (a-b) multiplied by itself three times: (a-b) * (a-b) * (a-b). This formula is essential for simplifying expressions and solving various mathematical problems.

### Expanding the Formula

The expanded form of (a-b)³ is:

**a³ - 3a²b + 3ab² - b³**

Let's break down how this formula is derived:

**First Expansion:**(a-b) * (a-b) = a² - 2ab + b²**Second Expansion:**(a² - 2ab + b²) * (a-b) = a³ - 3a²b + 3ab² - b³

### Applying the Formula

You can use the (a-b)³ formula to simplify expressions and solve equations. Here are some examples:

**Example 1:** Simplify (x-2)³

Using the formula, we get:

(x-2)³ = x³ - 3(x²)(2) + 3(x)(2²) - 2³
= **x³ - 6x² + 12x - 8**

**Example 2:** Solve for 'x' in the equation (x-1)³ = 8

**Expand the left side:**x³ - 3x² + 3x - 1 = 8**Move all terms to one side:**x³ - 3x² + 3x - 9 = 0**Factor the equation:**(x-3)(x² + 3) = 0**Solve for x:**x = 3 or x² = -3. Since the square of a real number cannot be negative, the only solution is**x = 3**.

### Remembering the Formula

The (a-b)³ formula can be easily remembered by following this pattern:

**First term:**Cube of the first term (a³)**Second term:**Three times the product of the square of the first term and the second term (3a²b)**Third term:**Three times the product of the first term and the square of the second term (3ab²)**Fourth term:**Cube of the second term (b³)

**Important Note:** The signs alternate between positive and negative in the expanded form of the formula.

By understanding and applying the (a-b)³ formula, you can simplify expressions, solve equations, and enhance your understanding of algebraic manipulations.