Express (4x + 8x^(2/3)) / (7√x) In The Form Ax^b + Cx^d

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Introduction

In this article, we will delve into the process of expressing the given function, (4x + 8x^(2/3)) / (7√x), in the form of ax^b + cx^d. This involves simplifying the expression using the properties of exponents and algebraic manipulations. Understanding how to rewrite functions in different forms is a fundamental skill in mathematics, particularly in calculus and algebra. This specific form, ax^b + cx^d, is a polynomial-like structure that makes it easier to analyze the behavior of the function, find its derivatives, and perform other mathematical operations. Before we proceed, it's essential to have a strong grasp of exponent rules, such as the quotient rule, power rule, and the relationship between fractional exponents and radicals. These rules will be our primary tools in transforming the given expression into the desired form. This exercise is not just about manipulating symbols; it's about gaining a deeper insight into the nature of functions and their representations. By the end of this article, you will not only be able to solve this particular problem but also apply the same techniques to a wide range of similar problems.

Understanding the Given Expression

To effectively express the function in the desired form, we must first thoroughly understand the given expression: (4x + 8x^(2/3)) / (7√x). This expression consists of two terms in the numerator, 4x and 8x^(2/3), and a single term in the denominator, 7√x. The presence of fractional exponents and radicals indicates that we will need to utilize the rules of exponents and radicals to simplify the expression. The term x^(2/3) represents the cube root of x squared, while √x is the square root of x. Recognizing these components is crucial for the subsequent steps. The overall goal is to separate the fraction into individual terms and then simplify each term by applying the appropriate exponent rules. This involves dividing each term in the numerator by the denominator and then combining like terms if necessary. The constants (4, 8, and 7) will also play a role in determining the coefficients a and c in the final form ax^b + cx^d. By carefully analyzing each component of the expression, we can develop a strategic approach to simplify it and rewrite it in the required form. This initial analysis is a critical step in problem-solving, as it lays the foundation for the subsequent algebraic manipulations.

Step-by-Step Solution

  1. Rewrite the square root: The first step in simplifying the expression (4x + 8x^(2/3)) / (7√x) is to rewrite the square root in the denominator using its exponential form. Recall that √x is equivalent to x^(1/2). Therefore, we can rewrite the expression as: (4x + 8x^(2/3)) / (7x^(1/2)). This transformation is crucial because it allows us to apply the rules of exponents more easily. By expressing the radical as a fractional exponent, we can combine terms with the same base by adding or subtracting their exponents, depending on whether they are in the numerator or denominator.

  2. Separate the fraction: Next, we separate the fraction into two individual terms by dividing each term in the numerator by the denominator: (4x / (7x^(1/2))) + (8x^(2/3) / (7x^(1/2))). This separation allows us to simplify each term independently, making the overall simplification process more manageable. Each term now represents a separate expression that can be simplified using the quotient rule of exponents.

  3. Apply the quotient rule of exponents: The quotient rule of exponents states that x^m / x^n = x^(m-n). We apply this rule to both terms. For the first term, we have 4x / (7x^(1/2)), which simplifies to (4/7)x^(1 - 1/2) = (4/7)x^(1/2). For the second term, we have 8x^(2/3) / (7x^(1/2)), which simplifies to (8/7)x^(2/3 - 1/2). To subtract the exponents, we need a common denominator, so we rewrite 2/3 - 1/2 as 4/6 - 3/6 = 1/6. Thus, the second term simplifies to (8/7)x^(1/6). Applying the quotient rule allows us to combine the exponents and express each term as a constant multiplied by a power of x.

  4. Combine the simplified terms: Now, we combine the simplified terms: (4/7)x^(1/2) + (8/7)x^(1/6). This is the expression in the form ax^b + cx^d, where a = 4/7, b = 1/2, c = 8/7, and d = 1/6. The expression is now in the desired form, with each term clearly showing the coefficient and the exponent of x. This form is much easier to work with for various mathematical operations, such as differentiation and integration.

Final Answer

Therefore, the expression (4x + 8x^(2/3)) / (7√x) can be expressed in the form ax^b + cx^d as:

(4/7)x^(1/2) + (8/7)x^(1/6).

This final answer clearly presents the function in the required format, making it easier to analyze and use in further mathematical computations. The process of arriving at this answer involved several key steps, including rewriting radicals as fractional exponents, separating the fraction into individual terms, applying the quotient rule of exponents, and combining the simplified terms. Each step was crucial in transforming the original expression into the desired form. This solution demonstrates the importance of understanding and applying the rules of exponents and radicals in simplifying algebraic expressions.

Importance of Understanding Exponent Rules

Understanding exponent rules is absolutely crucial for simplifying algebraic expressions and solving mathematical problems. In the context of the given problem, the ability to manipulate exponents allowed us to rewrite the expression (4x + 8x^(2/3)) / (7√x) into the form ax^b + cx^d. Without a solid grasp of these rules, the simplification process would be significantly more challenging, if not impossible. For instance, the quotient rule of exponents (x^m / x^n = x^(m-n)) was instrumental in combining terms with the same base. This rule allowed us to subtract the exponents when dividing terms, which was essential for simplifying the expression. Similarly, the ability to rewrite radicals as fractional exponents (e.g., √x = x^(1/2)) was crucial for applying the exponent rules effectively. Fractional exponents can initially seem daunting, but they are a powerful tool for simplifying expressions involving radicals. Furthermore, understanding how to work with negative exponents and zero exponents is also vital in more complex scenarios. Negative exponents indicate reciprocals, while a zero exponent results in 1 (except for 0^0, which is undefined). Mastering exponent rules not only simplifies algebraic manipulations but also provides a foundation for more advanced topics in mathematics, such as calculus and differential equations. These rules are the bedrock of algebraic simplification and are essential for anyone pursuing studies in mathematics, science, or engineering. In practical applications, exponent rules are used in various fields, including physics, computer science, and economics, to model and solve real-world problems. Therefore, investing time in understanding and practicing these rules is a worthwhile endeavor for anyone seeking to excel in quantitative fields.

Common Mistakes to Avoid

When simplifying expressions involving exponents and radicals, there are several common mistakes that students often make. Avoiding these mistakes is crucial for achieving accurate solutions. One frequent error is incorrectly applying the quotient rule of exponents. For example, students might mistakenly add the exponents instead of subtracting them when dividing terms with the same base. It's essential to remember that x^m / x^n = x^(m-n), not x^(m+n). Another common mistake is mishandling fractional exponents. Students may struggle to convert radicals to fractional exponents correctly or vice versa. Remember that √x = x^(1/2), ∛x = x^(1/3), and so on. A general rule is that the nth root of x is equivalent to x raised to the power of 1/n. Additionally, when dealing with expressions like x^(a/b), it's important to recognize that this represents the bth root of x raised to the power of a. Another error occurs when students fail to distribute exponents correctly. For instance, (xy)^n is equal to x^n * y^n, but many students incorrectly assume it is x^n * y. Similarly, when simplifying expressions with multiple terms, it's essential to follow the order of operations (PEMDAS/BODMAS) to avoid errors. This includes simplifying expressions within parentheses first, then exponents, multiplication and division, and finally addition and subtraction. Careless mistakes in arithmetic, such as incorrect addition or subtraction of fractions, can also lead to wrong answers. To mitigate these errors, it's helpful to double-check each step of the simplification process and to practice a variety of problems. Regular practice helps reinforce the correct application of exponent rules and reduces the likelihood of making these common mistakes. By being aware of these pitfalls and taking steps to avoid them, students can significantly improve their accuracy and proficiency in simplifying algebraic expressions.

Conclusion

In conclusion, expressing the function (4x + 8x^(2/3)) / (7√x) in the form ax^b + cx^d involves a systematic application of exponent rules and algebraic manipulations. The process begins with rewriting the square root as a fractional exponent, followed by separating the fraction into individual terms. The quotient rule of exponents is then applied to simplify each term, and finally, the simplified terms are combined to achieve the desired form. The final answer, (4/7)x^(1/2) + (8/7)x^(1/6), clearly demonstrates the function in the specified format. This exercise underscores the importance of a solid understanding of exponent rules and the ability to apply them correctly. These rules are fundamental in algebra and calculus and are essential for simplifying complex expressions. Moreover, it highlights the significance of avoiding common mistakes, such as misapplying the quotient rule or mishandling fractional exponents. Consistent practice and careful attention to detail are key to mastering these techniques. By working through this problem step-by-step, we have not only found the solution but also reinforced the underlying principles of algebraic manipulation. This skill is invaluable for tackling more advanced mathematical problems and is a cornerstone of mathematical proficiency. The ability to rewrite functions in different forms is a powerful tool in mathematical analysis, allowing us to gain deeper insights into their behavior and properties. Ultimately, the exercise of expressing functions in different forms is not just about finding the answer; it's about developing a deeper understanding of mathematical concepts and honing problem-solving skills.