Solving for (x^3 y^3) Given (xy 10) and (x^2 y^2 60): A Detailed Guide

In this article, we will walk through the process of solving a math problem involving polynomials and algebraic identities. Specifically, we are tasked with finding the value of x^3 y^3 given the equations xy 10 and x^2 y^2 60. To achieve this, we will use manipulation and algebraic identities to simplify the problem step by step.

Problem Type and Relevance

The given problem involves the manipulation of polynomials and algebraic expressions. Understanding how to solve such problems is crucial for students and professionals who work with advanced mathematics, data science, and engineering. It also brings to the forefront the importance of algebraic identities in simplifying complex expressions for easier computation.

Step-by-Step Solution

The first equation xy 10 directly gives us the product of (x) and (y).

xy 10

We need another piece of information, which is given by the second equation: x^2 y^2 60.

x^2 y^2 60

Step 1: Solve for xy

We start by using the identity that relates (x^2 y^2) to (xy): x^2 y^2 (xy)^2. Given that x^2 y^2 60, we can write:

(xy)^2 60

Since (xy 10), we substitute to get:

10^2 60 - 2xy

Rearranging for (xy), we get:

100 60 2xy 40 2xy xy 20

Step 2: Substitute to Find (x^3 y^3)

Now we have the values of (xy 10) and (xy 20). We need to find (x^3 y^3). We use the identity:

x^3 y^3 (xy)x^2 - (xy)y^2

Simplifying the second term, we get:

x^3 y^3 (xy)(x^2 - y^2)

Using the previously known (x^2 y^2 60) and substituting (xy 20), we have:

x^2 - y^2 x^2 y^2 / (xy) 60 / 20 3

Substituting back, we get:

x^3 y^3 20 * 40 400

Step 3: Verification

To verify, we can also use an alternative method:

x^3 y^3 xy(x^2 - y^2) xy(x^2 - y^2) 20 * (60 - 20) 20 * 40 400

Thus, the value of x^3 y^3 is indeed 400.

Conclusion

We have shown step-by-step how to solve for x^3 y^3 using algebraic identities and the given equations. The key is to understand and apply these identities correctly to simplify the expressions.

The approach demonstrated here is a fundamental technique in algebra, applicable in various mathematical and scientific contexts. The steps and the verification process ensure that the solution is both accurate and easy to understand.

Keywords: Algebraic Identities, Math Problem Solving, Polynomials