Solving the Expression of (frac{x}{a})^3 (frac{y}{b})^3 (frac{z}{c})^3 for Given Values

This article delves into a specific problem in algebra, where we explore a methodical approach to solve a complex expression involving fractions and variables. The problem at hand is to find the value of (frac{x}{a})^3 (frac{y}{b})^3 (frac{z}{c})^3 given the values of x ab - c, y bc - a, and z ca - b. We will break down the solution step-by-step, demonstrating the importance of algebraic manipulation and the use of algebraic identities.

Step-by-Step Solution

Step 1: Compute the Individual Fractions

The first step involves simplifying the given expressions for x/a, y/b, and z/c:

Calculate x/a

x/a frac{ab - c}{a} b - c

Calculate y/b

y/b frac{bc - a}{b} c - a

Calculate z/c

z/c frac{ca - b}{c} a - b

Step 2: Substitute into the Expression

Using the results from the previous step, we substitute these values into the original expression:

(b - c)^3 (c - a)^3 (a - b)^3

Step 3: Utilize Algebraic Identities

To further simplify the expression, we use the identity for the sum of cubes:

u^3 v^3 w^3 - 3uvw u v w (u^2 v^2 w^2 - uv - vw - wu)

Let u b - c, v c - a, w a - b.

Calculate u v w

(b - c)(c - a)(a - b) 0

Since u v w 0, the expression simplifies to:

(b - c)^3 (c - a)^3 (a - b)^3 3(b - c)(c - a)(a - b)

Final Solution

Therefore, the final result is:

(frac{x}{a})^3 (frac{y}{b})^3 (frac{z}{c})^3 3(b - c)(c - a)(a - b)

Conclusion

The derived expression is simplified using algebraic manipulation and the properties of the sum of cubes. This approach highlights the importance of recognizing patterns and utilizing identities in solving complex algebraic expressions.

Additional Insights

The method demonstrated in this solution not only provides a clear and concise approach to solving the problem but also lays down a foundation for understanding similar problems involving algebraic expressions and fractions.

Key Takeaways

Algebraic manipulation is crucial in simplifying complex expressions. Identifying and applying algebraic identities such as the sum of cubes can significantly simplify the calculations. Understanding the relationships between given variables can lead to more straightforward solutions.

The problem discussed here is a prime example of how effectively combining basic algebraic principles can lead to elegant solutions.