%5E3.jpeg "(xy)^3")
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.