Rewriting Exponential Expressions Express (x^(1/3))(x^(2/3)) In The Form X^(?/6)

Introduction

In the realm of mathematics, manipulating exponential expressions is a fundamental skill. This article delves into the process of rewriting the expression (x^(1/3))(x^(2/3)) in the form x^(?/6). This exercise not only reinforces the rules of exponents but also highlights the importance of finding common denominators when dealing with fractional exponents. Mastering these concepts is crucial for simplifying complex algebraic expressions and solving equations involving radicals and exponents. Our focus will be on breaking down the expression step-by-step, ensuring a clear understanding of the underlying principles. This exploration will be beneficial for students learning about exponents, as well as anyone looking to refresh their knowledge of algebraic manipulations. We will cover the basic rules of exponents, how to apply them to this specific problem, and the rationale behind each step. By the end of this article, you will have a solid grasp of how to rewrite exponential expressions with fractional exponents, and you'll be equipped to tackle similar problems with confidence. The process involves understanding how to multiply terms with the same base and how to combine exponents with different denominators. We will also emphasize the importance of accuracy in each step to avoid common pitfalls in algebraic manipulations. So, let's embark on this mathematical journey and unravel the intricacies of exponential expressions.

Understanding the Basics of Exponents

Before we dive into the specifics of rewriting (x^(1/3))(x^(2/3)), it's essential to revisit the fundamental rules of exponents. An exponent indicates how many times a base number is multiplied by itself. For instance, x^n means x multiplied by itself n times. When dealing with fractional exponents, such as in our problem, we encounter roots and powers. A fractional exponent like 1/n represents the nth root of the base. For example, x^(1/2) is the square root of x, and x^(1/3) is the cube root of x. A fractional exponent like m/n can be interpreted as both a power and a root: x^(m/n) is the nth root of x raised to the power of m, or equivalently, x raised to the power of m then taking the nth root. The key rule we'll use in this problem is the product of powers rule, which states that when multiplying terms with the same base, we add the exponents: x^a * x^b = x^(a+b). This rule is the cornerstone of simplifying expressions with exponents. Understanding this rule is crucial because it allows us to combine terms and simplify expressions that might initially appear complex. We must also remember that to add fractions, they must have a common denominator. This is a simple but critical point that will allow us to rewrite and combine the exponents in our expression effectively. In the following sections, we will apply this rule and the concept of common denominators to rewrite the given expression in the desired form. These basic principles are not only essential for this specific problem but are also fundamental to a wide range of mathematical problems involving exponents and algebraic manipulations.

Step-by-Step Solution: Rewriting the Expression

Now, let's apply these principles to rewrite the expression (x^(1/3))(x^(2/3)) in the form x^(?/6). The first step is to apply the product of powers rule: x^a * x^b = x^(a+b). In our case, a is 1/3 and b is 2/3. So, we have:

(x^(1/3))(x^(2/3)) = x^((1/3) + (2/3)).

Next, we need to add the exponents 1/3 and 2/3. Since they already have a common denominator of 3, we can directly add the numerators:

1/3 + 2/3 = (1 + 2) / 3 = 3/3.

Therefore, our expression becomes:

x^((1/3) + (2/3)) = x^(3/3).

Now, we simplify the exponent 3/3, which equals 1:

x^(3/3) = x^1 = x.

However, the question asks us to write the expression in the form x^(?/6). To do this, we need to rewrite the exponent 1 as a fraction with a denominator of 6. We know that 1 can be written as any number divided by itself. So, we can write 1 as 6/6:

x = x^(6/6).

Thus, we have rewritten the original expression in the required form. This step-by-step solution highlights the importance of applying exponent rules correctly and the necessity of finding common denominators when adding fractions. By carefully following each step, we can transform the expression into the desired form, demonstrating a solid understanding of exponential operations. This methodical approach is crucial in solving more complex problems involving exponents and algebraic manipulations.

Converting to the Target Form: x^(?/6)

Our goal is to express the simplified form, x, as x raised to some power with a denominator of 6, which is x^(?/6). We've already established that x is equivalent to x^1. Now, the key is to rewrite the exponent 1 as a fraction with a denominator of 6. To achieve this, we simply need to express 1 as a fraction where the numerator and denominator are the same, and the denominator is 6. The fraction that represents 1 with a denominator of 6 is 6/6. Therefore, we can rewrite x as x^(6/6). This transformation is crucial because it directly addresses the requirement of expressing the answer in the form x^(?/6). The question mark (?) is now replaced by 6. This step might seem simple, but it demonstrates an understanding of how to manipulate fractions and exponents to fit a specific format. It reinforces the idea that a number can be expressed in multiple ways without changing its value. For example, 1, 2/2, 3/3, 6/6, and many other fractions are all equal. This flexibility is vital when working with exponents and algebraic expressions. By rewriting x as x^(6/6), we have successfully answered the question and demonstrated our ability to work with fractional exponents and express numbers in different forms. This skill is essential for solving more complex problems in algebra and calculus.

Final Answer and Conclusion

In conclusion, by applying the product of powers rule and rewriting the exponent with a common denominator, we have successfully transformed the expression (x^(1/3))(x^(2/3)) into the form x^(?/6). Our step-by-step solution involved adding the exponents, simplifying the result, and then expressing the final exponent with a denominator of 6. The final answer is x^(6/6). This exercise underscores the importance of understanding and applying the basic rules of exponents, as well as the ability to manipulate fractions. These skills are fundamental in algebra and are essential for solving a wide range of mathematical problems. The process of rewriting the expression not only provides the correct answer but also reinforces the underlying mathematical principles. By breaking down the problem into smaller, manageable steps, we can clearly see how each rule and concept contributes to the final solution. This methodical approach is crucial for problem-solving in mathematics and other fields. Furthermore, this exercise highlights the versatility of numbers and expressions, as we demonstrated by rewriting 1 as 6/6 without changing its value. This flexibility is a key aspect of mathematical thinking and problem-solving. We hope this detailed explanation has provided a clear understanding of how to rewrite exponential expressions and has equipped you with the skills to tackle similar problems with confidence. Remember, practice is key to mastering these concepts, so continue to explore and challenge yourself with different types of exponential expressions and algebraic manipulations.