Expanding And Simplifying (x^(-2/3) - Y^(-1/2))^2 A Detailed Guide
In this detailed mathematical exploration, we will delve into the process of expanding and simplifying the algebraic expression (x^{-2/3} - y{-1/2})2. This expression involves fractional exponents and negative exponents, requiring a careful application of algebraic rules and exponent properties. Our journey will encompass understanding the fundamental concepts, applying the binomial square formula, dealing with negative and fractional exponents, and ultimately presenting the simplified form of the expression. This comprehensive guide is designed for students, educators, and anyone keen on honing their algebraic manipulation skills.
The expression (x^{-2/3} - y{-1/2})2 presents an interesting challenge due to the presence of negative and fractional exponents. To effectively expand and simplify this expression, we need to recall and apply several key algebraic principles. Firstly, we'll use the binomial square formula, which states that (a - b)^2 = a^2 - 2ab + b^2. Secondly, we'll leverage the properties of exponents, including how to handle negative exponents (a^{-n} = 1/a^n) and fractional exponents (a^{m/n} = \sqrt[n]{a^m}). By systematically applying these rules, we can transform the original expression into a more manageable and simplified form. Understanding these foundational concepts is crucial for not only solving this specific problem but also for tackling a wide range of algebraic challenges involving exponents and radicals. Let’s embark on this step-by-step simplification journey to master the techniques involved.
Applying the Binomial Square Formula
To initiate the simplification process of the expression **(x^-2/3} - y{-1/2})2**, the first crucial step involves applying the binomial square formula. This formula provides a direct method for expanding expressions in the form of (a - b)^2. The binomial square formula is mathematically expressed as and 'b' as y^{-1/2}. By substituting these values into the binomial square formula, we can expand the given expression into a trinomial, which we can then further simplify.
Applying the binomial square formula to **(x^-2/3} - y{-1/2})2** involves substituting x^{-2/3} for 'a' and y^{-1/2} for 'b' in the formula (a - b)^2 = a^2 - 2ab + b^2. This substitution yields the following expansion)^2 - 2(x{-2/3})(y{-1/2}) + (y{-1/2})2. This expanded form reveals the individual terms that we need to simplify further. Each term now involves powers of x and y, some of which are negative and fractional. The next step in our simplification process will focus on addressing these exponents using the properties of exponents. By carefully applying these properties, we can rewrite each term in a more standard form, making the entire expression easier to understand and manipulate. This step sets the stage for combining like terms and arriving at the final simplified form.
Simplifying Terms with Exponent Rules
After applying the binomial square formula, we arrive at the expanded form: (x{-2/3})2 - 2(x{-2/3})(y{-1/2}) + (y{-1/2})2. This expression now consists of three terms, each involving exponents. The next crucial step in simplifying this expression involves applying the fundamental rules of exponents. Specifically, we will use the power of a power rule, which states that (am)n = a^{m*n}, and the product of powers rule, which indirectly helps in handling terms like 2(x{-2/3})(y{-1/2}). These rules allow us to combine and simplify exponents, which is essential for reducing the complexity of the expression. By meticulously applying these rules, we can transform the terms into a more manageable form.
Let’s break down the simplification process term by term. For the first term, (x{-2/3})2, we apply the power of a power rule: (x{-2/3})2 = x^(-2/3)*2} = x^{-4/3}. This transformation shows how the exponent outside the parentheses multiplies with the exponent inside. Next, we consider the second term, -2(x{-2/3})(y{-1/2}). This term already has its simplest form in terms of exponent rules application but represents the product of the variables with their respective negative fractional exponents. Finally, for the third term, (y{-1/2})2, we again apply the power of a power rule)^2 = y^{(-1/2)*2} = y^{-1}. This systematic application of exponent rules is a fundamental skill in algebra, allowing us to manipulate and simplify complex expressions. By simplifying each term individually, we are now closer to the final simplified form of the original expression.
Handling Negative and Fractional Exponents
Having simplified the exponents in the expanded expression, we now have x^{-4/3} - 2x{-2/3}y{-1/2} + y^{-1}. This form still contains negative exponents and fractional exponents, which can be further addressed to present the expression in a more conventional and understandable format. To handle negative exponents, we'll use the rule a^{-n} = 1/a^n, which allows us to rewrite terms with negative exponents as fractions. For fractional exponents, we'll use the rule a^{m/n} = \sqrt[n]{a^m}, which relates fractional exponents to radicals (roots). By applying these rules, we can eliminate negative exponents and express fractional exponents as radicals, leading to a more simplified and readable expression.
Let's apply these rules to each term in our expression. The first term, x^{-4/3}, can be rewritten using the negative exponent rule as 1/x^{4/3}. Then, applying the fractional exponent rule, we can express x^{4/3} as \sqrt[3]{x^4}. Thus, x^{-4/3} becomes 1/\sqrt[3]{x^4}. Similarly, the third term, y^{-1}, can be rewritten as 1/y using the negative exponent rule. The middle term, -2x{-2/3}y{-1/2}, can be addressed in the same manner. We rewrite x^{-2/3} as 1/x^{2/3} and then as 1/\sqrt[3]{x^2}. Likewise, y^{-1/2} is rewritten as 1/y^{1/2} and then as 1/\sqrt{y}. Thus, the middle term becomes -2(1/\sqrt[3]{x^2})(1/\sqrt{y}), which simplifies to -2/(\sqrt[3]{x^2} * \sqrt{y}). By systematically converting negative and fractional exponents, we are transforming the expression into a form that is often preferred in mathematical representations.
Combining and Presenting the Simplified Expression
After addressing the negative and fractional exponents, our expression now takes the form 1/\sqrt[3]{x^4} - 2/(\sqrt[3]{x^2} * \sqrt{y}) + 1/y. This representation is significantly simplified compared to the original expression. However, it still involves fractions and radicals, which can be further manipulated for clarity and conciseness. To present the expression in its most simplified form, we can consider combining the terms over a common denominator. This process involves finding the least common multiple (LCM) of the denominators and rewriting each fraction with this common denominator. Alternatively, we can leave the expression as is, recognizing that it is a valid and simplified form.
In this case, combining the terms over a common denominator would involve finding the LCM of \sqrt[3]{x^4}, (\sqrt[3]{x^2} * \sqrt{y}), and y. This can be a complex process, and whether it leads to a “simpler” form is subjective and depends on the context. For many purposes, the expression 1/\sqrt[3]{x^4} - 2/(\sqrt[3]{x^2} * \sqrt{y}) + 1/y is considered sufficiently simplified. It clearly shows the relationships between x and y, and the use of radicals accurately represents the fractional exponents. Therefore, we can present this as the final simplified form of the original expression (x^{-2/3} - y{-1/2})2. This process demonstrates how algebraic expressions can be manipulated and simplified using a combination of algebraic rules and exponent properties. Understanding these techniques is crucial for advanced mathematical studies and applications.
In conclusion, we have successfully expanded and simplified the algebraic expression (x^{-2/3} - y{-1/2})2 through a series of methodical steps. We began by applying the binomial square formula, which allowed us to expand the expression into a trinomial. Following this, we employed the rules of exponents to simplify terms involving powers, particularly focusing on handling negative and fractional exponents. We transformed negative exponents into fractions and expressed fractional exponents as radicals, leading to a more conventional representation. Finally, we presented the simplified expression as 1/\sqrt[3]{x^4} - 2/(\sqrt[3]{x^2} * \sqrt{y}) + 1/y, which clearly demonstrates the relationships between x and y.
This detailed exploration has highlighted the importance of understanding and applying algebraic rules and exponent properties. The process involved not only mathematical manipulation but also a careful consideration of how to best represent the simplified form. While combining terms over a common denominator could be an option, we recognized that the current form provides a clear and concise representation of the expression. This exercise serves as a valuable example of how algebraic simplification is a fundamental skill in mathematics, with applications in various fields of study and real-world problem-solving. By mastering these techniques, students and practitioners can confidently tackle complex algebraic challenges and gain a deeper appreciation for the elegance and power of mathematical expressions.
