Simplifying Exponential Expressions A Step By Step Guide
In the realm of mathematics, simplifying expressions is a fundamental skill. Exponential expressions, with their powers and bases, often require careful manipulation to arrive at the most concise form. This article will delve into the process of simplifying a specific exponential expression, providing a clear, step-by-step explanation to enhance understanding. We will focus on the expression 3^(11/5) ÷ 3^(-1/5), breaking down each step and the underlying principles involved.
Understanding the Expression: Exponential Foundations
To effectively simplify the expression, it's crucial to grasp the fundamental concepts of exponents. At its core, an exponent indicates how many times a base number is multiplied by itself. For instance, in the expression 3^2, 3 is the base and 2 is the exponent, signifying 3 multiplied by itself (3 * 3 = 9). When dealing with fractional exponents, such as those in our expression, we venture into the realm of 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. Furthermore, a fractional exponent like m/n can be interpreted as taking the nth root of the base raised to the power of m, or equivalently, raising the nth root of the base to the power of m. This understanding forms the bedrock for simplifying expressions with fractional exponents.
In our specific expression, 3^(11/5) ÷ 3^(-1/5), we encounter two terms: 3^(11/5) and 3^(-1/5). The first term, 3^(11/5), signifies the fifth root of 3 raised to the power of 11. The second term, 3^(-1/5), introduces the concept of negative exponents. A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. In other words, x^(-n) is equivalent to 1/(x^n). Therefore, 3^(-1/5) represents the reciprocal of the fifth root of 3. Recognizing these fundamental interpretations of fractional and negative exponents is the first step towards simplifying the given expression. The ability to translate these abstract notations into concrete operations is essential for navigating the simplification process.
The Quotient Rule: Dividing Powers with the Same Base
When simplifying expressions involving division of powers with the same base, the quotient rule comes into play. This rule states that when dividing exponential expressions with the same base, you subtract the exponents. Mathematically, this is expressed as: a^m ÷ a^n = a^(m - n). This rule is a direct consequence of the definition of exponents and the properties of division. To illustrate, consider dividing x^5 by x^2. x^5 represents x multiplied by itself five times (x * x * x * x * x), and x^2 represents x multiplied by itself twice (x * x). When dividing x^5 by x^2, two of the x factors in the numerator cancel out with the two x factors in the denominator, leaving us with x multiplied by itself three times (x * x * x), which is x^3. This result aligns with the quotient rule, as 5 - 2 = 3.
The quotient rule provides a powerful shortcut for simplifying expressions involving division of powers. Instead of expanding the exponential terms and performing cancellations, we can directly subtract the exponents. This significantly streamlines the simplification process, especially when dealing with large or fractional exponents. The rule is applicable to any base (except 0) and any real number exponents. It is a cornerstone of exponential manipulation and a key tool in simplifying complex expressions. Mastering the quotient rule is essential for anyone working with exponents and algebraic simplification.
Applying the Quotient Rule: Step-by-Step Simplification
Now, let's apply the quotient rule to our expression: 3^(11/5) ÷ 3^(-1/5). As we've established, the quotient rule dictates that when dividing powers with the same base, we subtract the exponents. In this case, the base is 3, and the exponents are 11/5 and -1/5. Therefore, we can rewrite the expression as: 3^((11/5) - (-1/5)). The next step involves simplifying the exponent. We are subtracting a negative fraction, which is equivalent to adding the positive fraction. So, (11/5) - (-1/5) becomes (11/5) + (1/5). Adding these fractions, we get (11 + 1)/5, which simplifies to 12/5. Consequently, our expression now becomes 3^(12/5). This is a simplified form, but we can further explore it if needed.
To further clarify, we can express 3^(12/5) in radical form. Recall that a fractional exponent m/n represents the nth root of the base raised to the power of m. Thus, 3^(12/5) is the same as the fifth root of 3 raised to the power of 12, or (⁵√3)^12. Alternatively, it can be seen as the fifth root of 3^12, written as ⁵√(3^12). While 3^(12/5) is a valid and simplified exponential form, expressing it in radical form can sometimes offer additional insights or facilitate further simplification depending on the context. For instance, if we were to evaluate this expression numerically, the radical form might be more amenable to calculation using some calculators or software. The simplified exponential form, 3^(12/5), remains the most concise and mathematically standard representation of the original expression. The application of the quotient rule has allowed us to efficiently reduce a division of exponential terms into a single exponential term, demonstrating the power of this fundamental rule.
Re-evaluating the Options: Finding the Correct Answer
Having simplified the expression 3^(11/5) ÷ 3^(-1/5) to 3^(12/5), we now need to re-evaluate the answer options provided. The options are:
A. 1/12 B. 1/81 C. 81 D. 12
Comparing our simplified result, 3^(12/5), with the options, we can immediately see that none of the options directly match our result. The options are all integers or simple fractions, while our result is an exponential expression with a fractional exponent. This suggests that we might need to further analyze our simplified expression or consider whether there was a potential error in the options provided. Since 3^(12/5) is not equivalent to any of the given choices, let's revisit the simplification process to ensure accuracy.
Looking back at the application of the quotient rule, we subtracted the exponents: (11/5) - (-1/5) = (11/5) + (1/5) = 12/5. This part of the calculation seems correct. Therefore, 3^(12/5) is indeed the simplified form of the original expression. Now, considering the options again, we can conclude that there is likely an error in the provided choices. None of the options (1/12, 1/81, 81, 12) are equivalent to 3^(12/5). To further confirm this, we can approximate the value of 3^(12/5). We know that 3^(12/5) is the fifth root of 3 raised to the power of 12. This value is significantly larger than any of the provided options. Therefore, the correct answer is that none of the provided options are correct. The simplified form of the expression is 3^(12/5), which is not represented in the given choices.
Conclusion: Mastering Exponential Simplification
In this article, we've meticulously simplified the exponential expression 3^(11/5) ÷ 3^(-1/5). We began by understanding the fundamental concepts of exponents, including fractional and negative exponents. We then applied the quotient rule, a cornerstone of exponential simplification, which allows us to divide powers with the same base by subtracting their exponents. Through step-by-step calculations, we arrived at the simplified form, 3^(12/5). Finally, we compared our result with the provided answer options and concluded that none of the options were correct, highlighting the importance of accurate calculations and critical evaluation of results.
This exercise demonstrates the importance of a solid understanding of exponential rules and the ability to apply them systematically. The quotient rule, in particular, is a powerful tool for simplifying expressions involving division of powers. By mastering these concepts and techniques, you can confidently tackle a wide range of exponential simplification problems. Moreover, this process underscores the need to carefully review and validate results, ensuring that the final answer aligns with the mathematical principles and the context of the problem. Continued practice and application of these principles will solidify your understanding and proficiency in simplifying exponential expressions. This will enable you to approach more complex mathematical problems with greater confidence and accuracy.