Solving for a^3b^3 When (frac{a}{b}cdot frac{b}{a} 1)
In this article, we will explore how to find the value of (a^3b^3) given that (frac{a}{b} cdot frac{b}{a} 1). This problem requires a deep understanding of algebraic expressions and the application of trigonometric identities, making it particularly useful for students and professionals in mathematics and related fields.
Introduction
The given condition (frac{a}{b} cdot frac{b}{a} 1) is a fundamental algebraic identity. This simplification step forms the basis for solving more complex algebraic expressions. Let's delve into the process step-by-step.
Step-by-Step Solution
Step 1: Simplify the Given Equation
First, let's start by simplifying the given equation:
[ frac{a}{b} cdot frac{b}{a} 1 ]
Multiplying the terms on the left side gives:
(frac{a^2b^2}{ab} 1)
Simplifying the fraction:
(a^2b^2 ab)
Step 2: Apply the Sum of Cubes Formula
[ a^3 b^3 (a b)(a^2 - ab b^2) ]
However, we can also use the identity:
[ a^3b^3 a cdot b cdot (a^2 - ab b^2) ]
Step 3: Use the Simplified Equation
From the previous step, we know:
(a^2 - ab b^2 0)
Substituting this into the sum of cubes formula:
(a^3b^3 a cdot b cdot 0 0)
Thus, the value of (a^3b^3) is:
(0)
Conclusion
By following these steps, we have shown that when (frac{a}{b} cdot frac{b}{a} 1), the value of (a^3b^3) is (0).
Additional Notes
In the context of algebraic expressions, this problem showcases the importance of simplification and the use of identities in solving complex equations. Such problems are not only theoretical but also have practical implications in fields such as engineering, physics, and computer science.
References
1. Math is Fun: Basic Algebra 2. Khan Academy: Sum of Cubes 3. BetterExplained: A Visual Guide to Imaginary Numbers