## Understanding (xy)^3

In mathematics, the expression (xy)^3 represents the **cube of the product of x and y**. Here's a breakdown of what this means and how it works:

**What is a Cube?**

A cube of a number is the result of multiplying that number by itself three times. For example, the cube of 2 is 2 * 2 * 2 = 8.

**Applying it to (xy)^3**

In (xy)^3, the base of the exponent is the entire product "xy". So, we are cubing the entire product:

**(xy)^3 = (xy) * (xy) * (xy)**

**Simplifying the Expression**

To simplify the expression, we can use the commutative and associative properties of multiplication:

**(xy) * (xy) * (xy) = x * y * x * y * x * y**

Rearranging the terms:

**x * x * x * y * y * y = x^3 * y^3**

Therefore, **(xy)^3 = x^3 * y^3**.

**Key Point**

It's crucial to understand that cubing a product is not the same as cubing each factor individually and then multiplying the results. **(xy)^3 ≠ x^3 * y^3** This is a common mistake.

**Example**

Let's say x = 2 and y = 3.

- (xy)^3 = (2 * 3)^3 = 6^3 = 6 * 6 * 6 = 216
- x^3 * y^3 = 2^3 * 3^3 = 8 * 27 = 216

We can see that the results are the same, confirming our earlier simplification.

**In Conclusion**

Understanding how to work with exponents and products is fundamental in mathematics. The expression (xy)^3 highlights the importance of applying the correct order of operations and recognizing the difference between cubing a product and cubing individual factors.