Simplify The Expression: 1/(√2 + 1) * 1/(√2-1)

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In the realm of mathematics, simplifying expressions is a fundamental skill. It not only makes the expression easier to understand but also facilitates further calculations and analysis. In this article, we will delve into the simplification of the expression 1/(√2 1) * 1/(√2-1), providing a step-by-step guide to unravel its complexity. This process will involve techniques such as rationalizing the denominator, which is a crucial method in simplifying expressions involving radicals. By the end of this guide, you will not only be able to simplify this specific expression but also gain a deeper understanding of the underlying principles and techniques applicable to a broader range of mathematical problems.

Understanding the Expression

Before we embark on the simplification journey, let's first dissect the expression 1/(√2 1) * 1/(√2-1). This expression involves two fractions, each with a radical (√2) in the denominator. The presence of radicals in the denominator often complicates calculations and obscures the true value of the expression. Therefore, our primary goal is to eliminate these radicals from the denominator, a process known as rationalizing the denominator. This technique allows us to transform the expression into an equivalent form that is easier to manipulate and interpret. This initial step is crucial in setting the stage for further simplification and arriving at the most concise and understandable form of the expression. Understanding the structure of the expression is the first step towards effectively simplifying it.

Rationalizing the Denominator

The key to simplifying this expression lies in the technique of rationalizing the denominator. This involves eliminating the radical from the denominator of a fraction. In our case, we have two fractions with radicals in the denominator: 1/(√2 1) and 1/(√2-1). To rationalize the denominator, we multiply both the numerator and denominator of each fraction by the conjugate of the denominator. The conjugate of √2 1 is √2-1, and the conjugate of √2-1 is √2 1. This process leverages the difference of squares identity, which states that (a b)(a-b) = a² - b². By multiplying by the conjugate, we effectively eliminate the radical from the denominator.

For the first fraction, 1/(√2 1), we multiply both the numerator and denominator by √2-1:

(1/(√2 1)) * ((√2-1)/(√2-1)) = (√2-1) / ((√2)² - 1²) = (√2-1) / (2-1) = √2-1

Similarly, for the second fraction, 1/(√2-1), we multiply both the numerator and denominator by √2 1:

(1/(√2-1)) * ((√2 1)/(√2 1)) = (√2 1) / ((√2)² - 1²) = (√2 1) / (2-1) = √2 1

Now, we have successfully rationalized the denominators of both fractions, transforming them into simpler forms. This step is critical in simplifying expressions with radicals in the denominator, as it allows us to perform further operations without the complexity of dealing with radicals in the denominator. Rationalizing the denominator is a fundamental technique in algebra and is widely used in various mathematical contexts.

Multiplying the Simplified Fractions

Now that we have rationalized the denominators, our expression has transformed from 1/(√2 1) * 1/(√2-1) to (√2-1) * (√2 1). This step significantly simplifies the expression, making it easier to handle. We are now left with the task of multiplying two binomials, which can be done using the distributive property or the FOIL method (First, Outer, Inner, Last). This is a standard algebraic technique that allows us to expand the product of two binomials into a simpler form. Mastering this technique is crucial for simplifying various algebraic expressions and is a fundamental skill in algebra.

Applying the distributive property (or FOIL method), we multiply each term in the first binomial by each term in the second binomial:

(√2-1) * (√2 1) = (√2 * √2) (√2 * 1) (-1 * √2) (-1 * 1)

Simplifying each term, we get:

= 2 √2 - √2 - 1

Now, we combine like terms (√2 and -√2) to further simplify the expression:

= 2 - 1

This leads us to the final simplified form of the expression.

Final Simplification

After multiplying the simplified fractions, we arrived at the expression 2 - 1. This is a straightforward arithmetic operation that yields the final simplified result. Subtracting 1 from 2, we get:

2 - 1 = 1

Therefore, the simplified form of the original expression 1/(√2 1) * 1/(√2-1) is 1. This result demonstrates the power of rationalizing the denominator and applying basic algebraic techniques to simplify complex expressions. This final simplification step highlights the importance of performing all necessary arithmetic operations to arrive at the most concise and understandable form of the expression. The ability to simplify expressions to their simplest form is a fundamental skill in mathematics and is essential for solving more complex problems.

Conclusion

In this comprehensive guide, we have successfully simplified the expression 1/(√2 1) * 1/(√2-1) to its simplest form, which is 1. We achieved this by employing the technique of rationalizing the denominator, which involved multiplying the numerator and denominator of each fraction by its conjugate. This step eliminated the radicals from the denominator, making the expression easier to manipulate. We then multiplied the simplified fractions using the distributive property (or FOIL method) and combined like terms. Finally, we performed the necessary arithmetic operation to arrive at the final simplified result. This process not only demonstrates the specific steps involved in simplifying this particular expression but also illustrates the broader principles and techniques applicable to simplifying a wide range of mathematical expressions. The ability to simplify expressions is a crucial skill in mathematics, enabling us to solve problems more efficiently and gain a deeper understanding of mathematical concepts. By mastering these techniques, you can confidently tackle more complex mathematical challenges and appreciate the elegance and power of mathematical simplification.