Equivalent Expression For (1.5³ / 1.6⁴)⁻⁷ A Step-by-Step Solution
In the realm of mathematics, simplifying and manipulating expressions is a fundamental skill. One common task involves finding equivalent expressions, which are different forms that represent the same mathematical value. This article delves into the process of creating an equivalent expression for the given expression: (1.5³ / 1.6⁴)⁻⁷. We will explore the underlying principles of exponents, fractions, and negative exponents, and apply these concepts to arrive at the correct equivalent form. Understanding these concepts is crucial for any student studying algebra and pre-calculus, as they form the basis for more complex mathematical operations.
To tackle this problem, we need to recall the rules of exponents. Specifically, we'll focus on the following key rules:
- Power of a Quotient: (a/b)ⁿ = aⁿ / bⁿ
- Power of a Power: (aⁿ)ᵐ = aⁿᵐ
- Negative Exponent: a⁻ⁿ = 1/aⁿ
- Reciprocal of a Quotient: 1/(a/b) = b/a
These rules provide the foundation for manipulating the given expression and transforming it into an equivalent form. Let's break down the process step by step, applying these rules to unravel the complexities of the initial expression and reveal its equivalent counterpart. By meticulously applying each rule, we'll not only arrive at the solution but also gain a deeper appreciation for the elegance and consistency of mathematical principles. This process is not just about finding the answer; it's about understanding the why behind the answer, which is the hallmark of true mathematical understanding.
Let's start with the given expression: (1.5³ / 1.6⁴)⁻⁷. The presence of the negative exponent immediately suggests that we need to address this first. Recalling the rule for negative exponents, a⁻ⁿ = 1/aⁿ, we can rewrite the expression as:
1 / (1.5³ / 1.6⁴)⁷
This step transforms the negative exponent into a positive one, but it also introduces a fraction within a fraction. To simplify this, we can apply the rule for the reciprocal of a quotient, which states that 1/(a/b) = b/a. Applying this rule, we get:
(1.6⁴ / 1.5³ )⁷
Now, we have a simpler expression with a positive exponent. Next, we apply the power of a quotient rule, (a/b)ⁿ = aⁿ / bⁿ, to distribute the exponent 7 to both the numerator and the denominator:
- 6⁴ * ⁷ / 1.5³ * ⁷
This step separates the expression into two parts, each with its own exponent. The final step involves applying the power of a power rule, (aⁿ)ᵐ = aⁿᵐ, which tells us that when raising a power to another power, we multiply the exponents:
- 6^(47) / 1.5^(37)
Performing the multiplications in the exponents, we get:
- 6²⁸ / 1.5²¹
Therefore, the equivalent expression for (1.5³ / 1.6⁴)⁻⁷ is 1.6²⁸ / 1.5²¹. This step-by-step breakdown illustrates how the rules of exponents can be systematically applied to simplify complex expressions. The key is to identify the appropriate rule to apply at each stage and to execute the operations with precision. This methodical approach not only leads to the correct answer but also reinforces a deeper understanding of mathematical principles.
Now that we have simplified the expression to 1.6²⁸ / 1.5²¹, let's examine the given options and identify the one that matches our result:
A. 1. 5⁴ / 1.6³ B. 1. 6³ / 1.5⁴ C. 1. 5²¹ / 1.6²⁸ D. 1. 6²⁸ / 1.5²¹
By comparing our simplified expression with the options, it is clear that option D, 1. 6²⁸ / 1.5²¹, is the correct equivalent expression. The other options represent different manipulations of the original expression and do not match the result we obtained through the correct application of exponent rules. This exercise highlights the importance of careful and accurate simplification, as even a small error in applying the rules can lead to an incorrect answer. The process of matching our result with the available options reinforces the concept of equivalence, demonstrating that different forms of an expression can represent the same mathematical value.
When working with exponents, there are several common pitfalls that students often encounter. Understanding these potential errors and learning how to avoid them is crucial for success in algebra and beyond. One common mistake is misapplying the power of a power rule. For example, students might incorrectly add the exponents instead of multiplying them, resulting in errors like (aⁿ)ᵐ = aⁿ⁺ᵐ instead of the correct (aⁿ)ᵐ = aⁿᵐ. To avoid this, it's essential to memorize the rules accurately and practice applying them in various contexts. Another frequent error involves the negative exponent rule. Students might forget to take the reciprocal of the base, leading to incorrect simplifications. A helpful mnemonic for remembering this rule is that a negative exponent indicates a reciprocal. Finally, mistakes can arise from improper handling of fractions and the order of operations. Always remember to apply the exponent to both the numerator and the denominator when dealing with a fraction raised to a power, and adhere to the order of operations (PEMDAS/BODMAS) to ensure accurate simplification. By being aware of these common pitfalls and practicing diligently, students can significantly improve their ability to work with exponents and avoid costly errors.
The exponent rules are not just a set of abstract mathematical principles; they are fundamental tools that have far-reaching applications in various fields. In science, exponents are used extensively to express very large or very small numbers, such as the distance between stars or the size of an atom. The scientific notation, which relies heavily on exponents, provides a concise and manageable way to represent these values. In computer science, exponents play a crucial role in algorithms and data structures, particularly in areas like cryptography and data compression. The efficiency of many algorithms is often expressed in terms of exponential growth or decay. In finance, compound interest calculations rely on exponents to determine the future value of investments. The exponential growth of investments over time is a key concept in financial planning. Understanding exponent rules, therefore, is not just about solving mathematical problems; it's about developing a fundamental skill that is essential for success in a wide range of disciplines. By mastering these rules, students gain a powerful tool for analyzing and understanding the world around them.
In conclusion, finding equivalent expressions is a crucial skill in mathematics, and understanding exponent rules is paramount for this task. By systematically applying the rules of exponents, we successfully transformed the given expression (1.5³ / 1.6⁴)⁻⁷ into its equivalent form, 1. 6²⁸ / 1.5²¹. This process involved utilizing the power of a quotient rule, the power of a power rule, and the negative exponent rule. Recognizing and avoiding common pitfalls, such as misapplying the power of a power rule or neglecting the reciprocal in negative exponents, is essential for accurate simplification. The broader significance of exponent rules extends beyond mathematics, finding applications in science, computer science, finance, and various other fields. Mastering these rules equips individuals with a powerful tool for problem-solving and critical thinking in diverse contexts. Therefore, a solid understanding of exponents is not just a mathematical necessity but a valuable asset for navigating the complexities of the modern world.
