Understanding the Value of Complex Mathematical Expressions: 1/(2√3-2√2) * 3/(2√32√2)
This article delves into the solution of a complex mathematical expression, breaking down each step and elucidating the underlying principles. We will step through the process of simplifying the expression 1/(2√3-2√2) * 3/(2√32√2) and provide a detailed explanation of the mathematical steps involved. This content is suitable for students and professionals interested in mastering the intricacies of mathematics, particularly in the realm of sums and series and algebraic simplification.
Introduction
Mathematics, as a discipline, often presents seemingly daunting problems when simplified step by step and contextually explained. One such equation is the expression 1/(2√3-2√2) * 3/(2√32√2). This piece will guide you through the process of simplifying this expression, shedding light on the techniques used in algebra and simplifying complex fractions. Understanding these concepts is crucial for anyone looking to advance their knowledge in mathematics.
Step-by-Step Solution
Let's start by examining each part of the given expression.
Step 1: Simplify the Numerator
First, we simplify the numerator of the first fraction:
Numerator: 1/2√3-2√2
N2√3-2√2 * 3/ (2√3-2√2)
N8√3 - 4√2
N4(2√3 - √2)
The numerator is now simplified to 4(2√3 - √2).
Step 2: Simplify the Denominator
Next, we simplify the denominator of the overall expression:
D 2a2b / (2a - 2b)
D 2(2b)^2 / (2a - 2b)
D 4b^2 / 2(a - b)
D 2b^2 / (a - b)
Now, substitute the values 2√3 and 2√2 into the simplified form of the denominator:
D 8 / (2√3 - 1)
Step 3: Combine the Fractions
Now, we combine the simplified numerator and denominator:
N/D 4(2√3 - √2) / [8 / (2√3 - 1)]
N/D 4(2√3 - √2) * (2√3 - 1) / 8
N/D (8√3 - 4√2) * (2√3 - 1) / 8
Further simplifying the expression:
N/D (16√3 - 8√2 - 4√3 4√2) / 8
N/D (12√3 - 4√2) / 8
N/D (3√3 - √2) / 2
Thus, the final simplified form of the given expression is:
N/D (36√3 56 - 16√6 - 4√2) / 47
Conclusion
This detailed breakdown of a complex mathematical expression not only provides a clear solution but also illustrates the step-by-step process of simplification. The techniques used, such as rationalizing the denominator and breaking down fractions, are fundamental in advanced mathematics and are essential for problem-solving in sums and series. For a deeper understanding of these concepts, further reading on algebra and mathematical expressions is recommended.
Additional Resources
For those looking to learn more about simplifying complex expressions, consider exploring interactive online tutorials, supplementary textbooks on algebra, and practice problems to enhance your skills in this area.