Evaluating [(4/3)^-1 + (3/4)^2]^-1 A Step-by-Step Math Guide

Introduction

In this comprehensive guide, we will walk through the step-by-step evaluation of the mathematical expression [(4/3)^-1 + (3/4)^2]-1. This expression involves fractional exponents and requires a solid understanding of order of operations, fraction manipulation, and exponent rules. By breaking down each step, we will ensure clarity and accuracy in arriving at the final solution. This article aims to provide a detailed explanation suitable for students, educators, and anyone looking to refresh their algebra skills. Understanding how to simplify complex expressions is crucial in various fields, including mathematics, physics, engineering, and computer science. So, let's dive into the process and unravel this expression systematically.

Understanding the Basics

Before we start the evaluation, it's essential to grasp some fundamental concepts related to exponents and fractions. Understanding these basics will help in simplifying the expression [(4/3)^-1 + (3/4)^2]-1 more effectively.

First, let's discuss negative exponents. A negative exponent indicates the reciprocal of the base raised to the positive exponent. Mathematically, this is expressed as a^-n = 1/a^n. In our expression, we have (4/3)^-1, which means we need to take the reciprocal of 4/3. Understanding this rule is crucial because it transforms a negative exponent into a positive one, making it easier to work with the fraction. For instance, if we have 2^-1, it means 1/2^1, which simplifies to 1/2. Similarly, a more complex example like (2/3)^-2 would mean 1/(2/3)^2, which we will simplify further in later steps.

Next, let's talk about fractional exponents and their squares. When we square a fraction, we are essentially multiplying the fraction by itself. For example, (a/b)^2 means (a/b) * (a/b), which equals a^2/b2. In our expression, we have (3/4)^2, which means we need to square both the numerator and the denominator. Squaring a fraction is a straightforward process, but it’s important to do it correctly to avoid errors. For instance, (1/2)^2 would be (1/2) * (1/2), resulting in 1/4. Similarly, (5/6)^2 would be (5/6) * (5/6), which equals 25/36. These basic calculations form the building blocks for more complex operations.

Finally, the order of operations is a critical concept to keep in mind. We follow the PEMDAS/BODMAS rule, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order ensures that we perform the operations in the correct sequence, leading to the correct answer. In our expression, we first need to handle the exponents, then the addition within the brackets, and finally the exponent outside the brackets. Ignoring the order of operations can lead to completely different and incorrect results. For example, if we were to add before handling exponents, we would significantly alter the outcome. This is why adherence to PEMDAS/BODMAS is paramount in mathematical calculations.

By understanding these fundamental concepts – negative exponents, fractional exponents, and the order of operations – we are well-prepared to tackle the evaluation of the given expression. Each concept plays a vital role in simplifying the expression systematically and accurately. With these basics in mind, let’s move on to the step-by-step evaluation.

Step 1: Dealing with the Negative Exponent

The first step in evaluating the expression [(4/3)^-1 + (3/4)^2]-1 is to address the negative exponent. We have (4/3)^-1 inside the brackets. As discussed earlier, a negative exponent means we take the reciprocal of the base. Therefore, (4/3)^-1 is the same as 1/(4/3), which simplifies to 3/4.

To elaborate further, when you have a fraction raised to a negative exponent, you simply flip the fraction and change the sign of the exponent. In this case, (4/3)^-1 becomes (3/4)^1. Since any number raised to the power of 1 is the number itself, (3/4)^1 is just 3/4. This transformation makes the expression easier to work with because we’ve eliminated the negative exponent.

It's important to understand why this works. The rule a^-n = 1/a^n is a fundamental property of exponents. When applied to fractions, it means (a/b)^-n = (b/a)^n. This rule helps us to convert expressions with negative exponents into equivalent expressions with positive exponents, which are generally easier to handle. In our case, converting (4/3)^-1 to 3/4 is a straightforward application of this rule.

By addressing the negative exponent first, we simplify the expression inside the brackets, making the subsequent steps more manageable. This initial step is crucial because it sets the stage for the rest of the evaluation. If we were to skip this step or handle it incorrectly, it would impact the entire calculation. The reciprocal operation is a core concept in mathematics, and mastering it is essential for simplifying complex expressions.

Now that we have simplified (4/3)^-1 to 3/4, our expression becomes [(3/4) + (3/4)^2]-1. The next step involves dealing with the square of the fraction, which we will address in the following section. By methodically simplifying each part of the expression, we maintain clarity and reduce the chances of making errors. This step-by-step approach is key to successfully evaluating complex mathematical expressions.

Step 2: Evaluating the Square

Having addressed the negative exponent, the next step in evaluating [(4/3)^-1 + (3/4)^2]-1 is to evaluate the square of the fraction. We have (3/4)^2 inside the brackets. Squaring a fraction means multiplying the fraction by itself. So, (3/4)^2 is equal to (3/4) * (3/4).

To calculate this, we multiply the numerators together and the denominators together. Thus, (3/4) * (3/4) = (33) / (44) = 9/16. This operation is a straightforward application of the rules for multiplying fractions. The key here is to ensure that both the numerator and the denominator are squared correctly. Squaring each part individually helps maintain accuracy and avoids common mistakes.

Understanding how to square fractions is crucial because it appears frequently in various mathematical contexts, from algebraic equations to calculus problems. The process is always the same: multiply the numerator by itself and the denominator by itself. For example, if we had (2/5)^2, it would be (22) / (55) = 4/25. Similarly, (1/3)^2 would be (11) / (33) = 1/9. These simple examples illustrate the consistency of the rule.

Now that we've evaluated (3/4)^2 to be 9/16, our expression becomes [(3/4) + (9/16)]^-1. We have now simplified both the negative exponent and the squared fraction within the brackets. The expression is gradually becoming more manageable, and we are one step closer to the final solution. This step highlights the importance of breaking down a complex problem into smaller, more digestible parts.

By methodically addressing each exponent and operation, we are ensuring that our calculations are accurate. This step-by-step approach minimizes the chance of errors and allows us to focus on each individual component of the expression. Evaluating the square is a fundamental skill in algebra, and mastering it is essential for handling more complex mathematical problems.

In the next step, we will focus on adding the two fractions inside the brackets. This will involve finding a common denominator and combining the numerators. By continuing this systematic approach, we will arrive at the simplified expression and the final answer.

Step 3: Adding the Fractions

With the exponents dealt with, our expression [(4/3)^-1 + (3/4)^2]-1 has been simplified to [(3/4) + (9/16)]^-1. The next step involves adding the two fractions inside the brackets. To add fractions, they must have a common denominator. In this case, we need to find the least common multiple (LCM) of 4 and 16.

The least common multiple of 4 and 16 is 16. This means we need to convert 3/4 into an equivalent fraction with a denominator of 16. To do this, we multiply both the numerator and the denominator of 3/4 by 4. So, (3/4) * (4/4) = 12/16. Now, we can rewrite the expression inside the brackets as (12/16) + (9/16).

Understanding how to find a common denominator is a fundamental skill in fraction arithmetic. It allows us to combine fractions with different denominators into a single fraction, making further calculations possible. The LCM is the smallest number that both denominators can divide into evenly. In our case, 16 is the smallest number that both 4 and 16 can divide into.

Now that we have a common denominator, we can add the fractions by adding the numerators and keeping the denominator the same. So, (12/16) + (9/16) = (12 + 9) / 16 = 21/16. This gives us a single fraction inside the brackets, which is a significant step towards simplifying the overall expression. The addition of fractions is a basic operation, but it's crucial to perform it accurately to avoid errors in the final result.

The ability to add fractions efficiently and correctly is essential for various mathematical contexts. Whether you’re solving algebraic equations, dealing with ratios and proportions, or working on calculus problems, adding fractions is a foundational skill. The process involves identifying the common denominator, converting the fractions, and then adding the numerators.

Now that we have simplified the expression inside the brackets to 21/16, our overall expression becomes [21/16]^-1. We are now down to the final step, which involves dealing with the remaining exponent. This final step will give us the ultimate simplified answer to the original expression. By systematically working through each step, we have made the problem much more manageable and are confident in our calculations.

In the next section, we will address the final exponent and complete the evaluation of the expression. By carefully following the order of operations and applying the rules of exponents and fractions, we will arrive at the correct solution.

Step 4: Final Inversion

Having simplified the expression inside the brackets, we are now at the final step: inverting the fraction. Our expression currently stands as [21/16]^-1. As we discussed in the beginning, a negative exponent means we take the reciprocal of the base. In this case, we need to take the reciprocal of the fraction 21/16.

Taking the reciprocal of a fraction simply means flipping the numerator and the denominator. So, the reciprocal of 21/16 is 16/21. Therefore, [21/16]^-1 is equal to 16/21. This step is straightforward but crucial for completing the evaluation. The negative exponent effectively instructs us to invert the fraction, which is a fundamental operation in dealing with exponents and fractions.

Understanding reciprocals is essential in various areas of mathematics. Reciprocals are used in solving equations, simplifying expressions, and working with ratios and proportions. The reciprocal of a number is simply 1 divided by that number. For a fraction, the reciprocal is found by swapping the numerator and the denominator. For instance, the reciprocal of 3/5 is 5/3, and the reciprocal of 7 (which can be written as 7/1) is 1/7.

By taking the reciprocal of 21/16, we have completed the final step in evaluating the expression. Our journey from the original complex expression [(4/3)^-1 + (3/4)^2]-1 to the final simplified answer of 16/21 demonstrates the power of breaking down a problem into smaller, manageable steps. We have systematically addressed each exponent, performed the necessary arithmetic operations, and arrived at the solution with confidence.

The final answer, 16/21, is a fraction in its simplest form, as 16 and 21 have no common factors other than 1. This means that the fraction cannot be further reduced. The ability to simplify expressions to their simplest form is a key skill in mathematics, as it provides the most concise and clear representation of the answer.

In conclusion, we have successfully evaluated the expression [(4/3)^-1 + (3/4)^2]-1 by following a step-by-step approach. We started by addressing the negative exponent, then evaluated the square, added the resulting fractions, and finally, inverted the fraction. Each step was performed carefully and methodically, ensuring accuracy and clarity. This process highlights the importance of understanding the rules of exponents and fractions, as well as the order of operations, in simplifying complex mathematical expressions. The result, 16/21, is the final simplified answer.

Conclusion

In summary, we have meticulously evaluated the expression [(4/3)^-1 + (3/4)^2]-1, arriving at the simplified result of 16/21. This process involved several key steps, each building upon the previous one to systematically reduce the complexity of the expression. By addressing the negative exponent, evaluating the square, adding fractions, and taking the final reciprocal, we demonstrated a thorough understanding of exponent rules, fraction manipulation, and the order of operations.

Throughout this guide, we emphasized the importance of breaking down complex problems into smaller, more manageable parts. This approach not only makes the problem less daunting but also minimizes the risk of errors. Each step was explained in detail, ensuring that readers of all levels could follow along and grasp the underlying concepts. The initial step of dealing with the negative exponent transformed (4/3)^-1 into 3/4, setting the stage for subsequent calculations. Then, evaluating (3/4)^2 resulted in 9/16, which prepared us for the addition of fractions.

The addition of fractions required finding a common denominator, which is a fundamental skill in arithmetic. We converted 3/4 to 12/16 to match the denominator of 9/16, allowing us to add the fractions and obtain 21/16. This step highlighted the importance of understanding equivalent fractions and the least common multiple. Finally, taking the reciprocal of 21/16 gave us the ultimate simplified answer of 16/21.

This step-by-step evaluation not only provides a solution to the specific expression but also reinforces essential mathematical principles. The ability to work with exponents, fractions, and the order of operations is crucial for success in algebra and beyond. The methodical approach we employed can be applied to a wide range of mathematical problems, making it a valuable skill for students, educators, and anyone interested in mathematics.

The final result, 16/21, is a fraction in its simplest form, representing the exact value of the original expression. This example illustrates the beauty and precision of mathematics, where complex expressions can be simplified to elegant and concise forms. By mastering the techniques demonstrated in this guide, you can confidently tackle similar problems and enhance your mathematical proficiency.

In conclusion, the evaluation of [(4/3)^-1 + (3/4)^2]-1 to 16/21 serves as a practical demonstration of fundamental mathematical concepts. The step-by-step approach not only simplifies the problem but also reinforces the importance of understanding and applying basic rules of arithmetic and algebra. This guide provides a solid foundation for further exploration in mathematics and encourages a systematic approach to problem-solving.