Expressing (3x - 7x^3 + 7x^2 + X^(2/3)) / √x In The Form Ax^b + Cx^d + Ex^f + Gx^h

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When confronted with the task of expressing a complex algebraic expression in a simplified polynomial form, a systematic approach is crucial. In this article, we will dissect the expression (3x - 7x^3 + 7x^2 + x^(2/3)) / √x and transform it into the desired format of ax^b + cx^d + ex^f + gx^h. This process involves a combination of algebraic manipulation, exponent rules, and a keen eye for detail. The goal is to break down the expression into its constituent terms, each with a distinct coefficient and exponent, thereby revealing the underlying structure of the polynomial.

To begin, let's clearly state the expression we aim to simplify:

(3x - 7x^3 + 7x^2 + x^(2/3)) / √x

Our mission is to rewrite this expression in the form:

ax^b + cx^d + ex^f + gx^h

where a, c, e, and g are coefficients, and b, d, f, and h are exponents. This form is a standard representation of a polynomial, making it easier to analyze and manipulate.

This task is not merely an academic exercise. The ability to express polynomials in standard form is fundamental in various mathematical and scientific contexts. It allows for easier comparison of polynomials, simplification of equations, and application of polynomial functions in modeling real-world phenomena. From physics to engineering, the manipulation of polynomial expressions is a cornerstone of problem-solving. Understanding how to transform complex expressions into simpler forms enhances one's mathematical toolkit and provides a solid foundation for advanced studies.

In the subsequent sections, we will embark on a step-by-step journey, meticulously applying the rules of algebra and exponents to achieve our objective. We will first address the square root in the denominator, then distribute the division across the terms in the numerator, and finally, simplify each term to match the target format. By the end of this process, we will have successfully expressed the given expression in the desired polynomial form, showcasing the power and elegance of algebraic manipulation.

Step-by-Step Simplification

1. Rewrite the Square Root

Our initial step in simplifying the expression (3x - 7x^3 + 7x^2 + x^(2/3)) / √x involves addressing the square root in the denominator. The square root of x, denoted as √x, can be equivalently expressed using fractional exponents. Specifically, √x is the same as x raised to the power of 1/2, or x^(1/2). This transformation is a fundamental application of the relationship between radicals and exponents, which states that the nth root of a number is the same as raising that number to the power of 1/n. This conversion is not just a notational change; it sets the stage for applying the rules of exponents in the subsequent steps.

By rewriting the denominator, we transform the expression into a form that is more amenable to algebraic manipulation. The presence of a fractional exponent allows us to use the quotient rule of exponents, which is crucial for simplifying expressions involving division of terms with the same base. This rule states that when dividing powers with the same base, you subtract the exponents. By converting the square root to a fractional exponent, we unlock the potential to use this rule effectively.

In the context of polynomial simplification, dealing with radicals can often be cumbersome. Rewriting them as fractional exponents streamlines the process and allows for a more direct application of algebraic rules. This is a common technique in algebra and calculus, where simplifying expressions is often a prerequisite for further analysis or computation. For instance, when finding the derivative or integral of a function involving radicals, it is almost always beneficial to first convert them to fractional exponents.

By rewriting √x as x^(1/2), we lay the groundwork for the next step, which involves distributing the division across the terms in the numerator. This seemingly small change is a critical step in the overall simplification process, highlighting the importance of understanding and applying the basic rules of exponents and radicals.

2. Distribute the Division

Having rewritten the denominator as x^(1/2), our next task is to distribute the division across each term in the numerator. This process involves dividing each term in the numerator (3x, -7x^3, 7x^2, and x^(2/3)) by x^(1/2). This is a direct application of the distributive property of division over addition and subtraction, a fundamental concept in algebra. By performing this distribution, we break down the original expression into four simpler fractions, each of which can be simplified independently.

The significance of this step lies in its ability to transform a single complex fraction into a sum of simpler terms. This transformation is crucial because it allows us to apply the quotient rule of exponents to each term individually. The quotient rule, as mentioned earlier, states that when dividing powers with the same base, you subtract the exponents. By distributing the division, we set the stage for applying this rule to each term, leading to a significant simplification of the expression.

Consider the impact of this step on the overall simplification process. Instead of grappling with a single fraction containing multiple terms and exponents, we now have four separate fractions, each of which is easier to manage. This is a common strategy in algebra: breaking down complex problems into smaller, more manageable parts. By applying the distributive property, we make the problem more accessible and reduce the likelihood of errors.

The result of this distribution is a series of terms, each consisting of a coefficient and a power of x. These terms are now in a form that is much closer to our target format of ax^b + cx^d + ex^f + gx^h. The next step involves simplifying each of these terms by applying the quotient rule of exponents. This will reveal the specific values of the coefficients and exponents in our desired polynomial form.

3. Simplify Each Term

With the division distributed, we now have four individual terms to simplify: 3x / x^(1/2), -7x^3 / x^(1/2), 7x^2 / x^(1/2), and x^(2/3) / x^(1/2). The key to simplifying each of these terms lies in applying the quotient rule of exponents. This rule, a cornerstone of exponent manipulation, dictates that when dividing powers with the same base, you subtract the exponents. In mathematical notation, this is expressed as x^m / x^n = x^(m-n).

Let's apply this rule to each term systematically:

  • 3x / x^(1/2): Here, we have x^1 divided by x^(1/2). Subtracting the exponents gives us 1 - 1/2 = 1/2. Thus, this term simplifies to 3x^(1/2).
  • -7x^3 / x^(1/2): In this case, we have x^3 divided by x^(1/2). Subtracting the exponents yields 3 - 1/2 = 5/2. This term simplifies to -7x^(5/2).
  • 7x^2 / x^(1/2): Here, we have x^2 divided by x^(1/2). Subtracting the exponents gives us 2 - 1/2 = 3/2. This term simplifies to 7x^(3/2).
  • x^(2/3) / x^(1/2): This term involves fractional exponents in both the numerator and the denominator. Subtracting the exponents requires finding a common denominator for 2/3 and 1/2, which is 6. Thus, we have 2/3 - 1/2 = 4/6 - 3/6 = 1/6. This term simplifies to x^(1/6).

By applying the quotient rule to each term, we have successfully reduced the complexity of the expression. Each term now consists of a coefficient and a power of x, precisely the form we are aiming for. This step is a testament to the power of exponent rules in simplifying algebraic expressions. The seemingly complex division has been transformed into a series of simple subtractions, resulting in a much cleaner and more manageable expression.

4. Final Result

After meticulously simplifying each term, we have arrived at the final form of the expression. The original expression, (3x - 7x^3 + 7x^2 + x^(2/3)) / √x, has been transformed into a sum of terms, each with a distinct coefficient and exponent. This final form not only satisfies the initial requirement of expressing the polynomial in the form ax^b + cx^d + ex^f + gx^h but also provides a clear and concise representation of the polynomial's structure.

The simplified expression is:

3x^(1/2) - 7x^(5/2) + 7x^(3/2) + x^(1/6)

This result showcases the elegance of algebraic manipulation. By systematically applying the rules of exponents and the distributive property, we have successfully transformed a complex fraction into a more manageable polynomial form. Each term in the expression is now clearly defined, with the coefficients being 3, -7, 7, and 1, and the exponents being 1/2, 5/2, 3/2, and 1/6, respectively.

This final form is not just a cosmetic change; it has significant implications for further mathematical operations. For instance, if we were to differentiate or integrate this expression, the simplified form would make the process much more straightforward. Similarly, if we were to analyze the behavior of this polynomial function, the clear separation of terms would allow for a more intuitive understanding of its properties.

In summary, the process of expressing (3x - 7x^3 + 7x^2 + x^(2/3)) / √x in the form ax^b + cx^d + ex^f + gx^h has been a journey through the fundamental principles of algebra. From rewriting radicals as fractional exponents to distributing division and applying the quotient rule, each step has contributed to the final result. This exercise not only reinforces the importance of these algebraic techniques but also demonstrates the power of systematic problem-solving in mathematics.

Conclusion

In conclusion, the journey of expressing the given algebraic expression (3x - 7x^3 + 7x^2 + x^(2/3)) / √x in the desired form of ax^b + cx^d + ex^f + gx^h has been a comprehensive exercise in algebraic manipulation. We began by recognizing the need to address the square root in the denominator, transforming it into a fractional exponent to facilitate the application of exponent rules. This initial step was crucial in setting the stage for subsequent simplifications.

The distribution of division across each term in the numerator was a pivotal move, allowing us to break down the complex fraction into a series of simpler fractions. This application of the distributive property highlighted its importance in transforming complex expressions into more manageable components. Each resulting fraction then became the subject of simplification using the quotient rule of exponents, where we meticulously subtracted exponents to arrive at the desired form.

The final result, 3x^(1/2) - 7x^(5/2) + 7x^(3/2) + x^(1/6), stands as a testament to the power and elegance of algebraic techniques. It not only fulfills the initial requirement of expressing the polynomial in the specified form but also provides a clear and concise representation of the polynomial's structure. The coefficients and exponents are now explicitly defined, allowing for a deeper understanding of the polynomial's behavior and properties.

This exercise underscores the significance of mastering fundamental algebraic principles. The ability to manipulate expressions, apply exponent rules, and distribute operations is essential in various mathematical and scientific contexts. From simplifying equations to analyzing functions, these skills form the bedrock of problem-solving. Furthermore, this process highlights the importance of systematic thinking in mathematics. By breaking down a complex problem into smaller, more manageable steps, we can effectively navigate challenges and arrive at elegant solutions.

The transformation we have undertaken is not merely an academic exercise. It is a practical demonstration of the tools and techniques used in real-world applications of mathematics. Whether in physics, engineering, or computer science, the ability to simplify and manipulate algebraic expressions is a valuable asset. The skills honed in this exercise provide a solid foundation for tackling more advanced mathematical concepts and solving complex problems.

Ultimately, this journey through algebraic manipulation serves as a reminder of the beauty and power of mathematics. By applying logical rules and systematic techniques, we can unravel complexity and reveal the underlying structure of mathematical expressions. The final result is not just an answer; it is a testament to the elegance and efficiency of mathematical thought.