Evaluate (1/2)^3 - (1/2)^2

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In this article, we will delve into the process of evaluating the mathematical expression (1/2)^3 - (1/2)^2. This seemingly simple expression provides a great opportunity to reinforce fundamental concepts related to exponents, fractions, and order of operations. Understanding how to solve such expressions is crucial for building a solid foundation in algebra and beyond. We will break down the problem step-by-step, explaining each operation clearly and concisely. The concepts covered here are essential building blocks for more complex mathematical problems, making this a worthwhile exploration for students of all levels. This article aims to provide a thorough understanding of the concepts involved, ensuring that the reader can confidently tackle similar problems in the future.

Understanding Exponents

Before we dive into the specific calculation, let's revisit the concept of exponents. An exponent indicates how many times a base number is multiplied by itself. For instance, in the expression x^n, x is the base and n is the exponent. This means we multiply x by itself n times. In our problem, we encounter the terms (1/2)^3 and (1/2)^2. Understanding exponents is crucial for accurately solving this equation. The term (1/2)^3 signifies that we need to multiply 1/2 by itself three times, which can be written as (1/2) * (1/2) * (1/2). Similarly, (1/2)^2 means we multiply 1/2 by itself twice, or (1/2) * (1/2). Correctly interpreting the exponents is the first step towards arriving at the correct solution. This fundamental understanding lays the groundwork for more advanced algebraic manipulations and is a cornerstone of mathematical literacy. Grasping the concept of exponents not only helps in evaluating numerical expressions but also in simplifying algebraic expressions and solving equations. By carefully considering the base and the exponent, we can avoid common errors and ensure accurate calculations. Therefore, a firm grasp of exponents is paramount for success in mathematics.

Evaluating (1/2)^3

Let's first evaluate the term (1/2)^3. As we established earlier, this means we need to multiply 1/2 by itself three times: (1/2) * (1/2) * (1/2). When multiplying fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, (1/2) * (1/2) equals 1/4 (because 11 = 1 and 22 = 4). Now we need to multiply 1/4 by 1/2. Again, multiplying the numerators and denominators gives us (11)/(42) which simplifies to 1/8. Therefore, (1/2)^3 equals 1/8. This step-by-step breakdown is crucial for ensuring accuracy and understanding. It highlights the fundamental principles of fraction multiplication and exponent application. A clear grasp of these principles is essential for tackling more complex mathematical problems. By carefully breaking down each step, we minimize the risk of errors and build a solid foundation in mathematical computation. The ability to confidently evaluate expressions like (1/2)^3 is a key skill in algebra and beyond, enabling us to solve equations, simplify expressions, and approach more challenging mathematical concepts with assurance.

Evaluating (1/2)^2

Next, we need to evaluate the term (1/2)^2. This means we multiply 1/2 by itself twice: (1/2) * (1/2). As we discussed in the previous section, when multiplying fractions, we multiply the numerators together and the denominators together. So, (1/2) * (1/2) equals 1/4 (because 11 = 1 and 22 = 4). Therefore, (1/2)^2 equals 1/4. This calculation is straightforward but crucial for the overall solution. Understanding how to square a fraction is a fundamental skill in mathematics. It reinforces the concept of exponents and fraction multiplication, both of which are essential for more complex algebraic manipulations. This step-by-step approach ensures accuracy and helps to solidify the understanding of these basic operations. Mastering these fundamental skills is essential for building confidence in mathematics and for tackling more challenging problems in the future. By carefully evaluating (1/2)^2, we lay the groundwork for the final step of our calculation, which involves subtracting this value from the result of (1/2)^3.

Subtracting the Results

Now that we have evaluated both terms, (1/2)^3 = 1/8 and (1/2)^2 = 1/4, we can perform the subtraction: 1/8 - 1/4. To subtract fractions, they must have a common denominator. The least common denominator (LCD) of 8 and 4 is 8. We need to convert 1/4 into an equivalent fraction with a denominator of 8. To do this, we multiply both the numerator and the denominator of 1/4 by 2: (12)/(42) = 2/8. Now we have the subtraction problem as 1/8 - 2/8. Subtracting the numerators, we get 1 - 2 = -1. The denominator remains the same, so the result is -1/8. Therefore, (1/2)^3 - (1/2)^2 = -1/8. This final step demonstrates the importance of understanding fraction subtraction and the concept of common denominators. It brings together all the previous steps to arrive at the final answer. The ability to accurately subtract fractions is a crucial skill in mathematics, and this example reinforces that skill in the context of exponent evaluation. The negative result highlights the importance of paying attention to signs and the order of operations. By carefully following each step, we have successfully evaluated the given expression and arrived at the correct solution.

Final Answer

In conclusion, after carefully evaluating each term and performing the subtraction, we have found that (1/2)^3 - (1/2)^2 = -1/8. This problem serves as a valuable exercise in applying fundamental mathematical principles, including exponents, fraction multiplication, and subtraction of fractions with common denominators. By breaking down the problem into smaller, manageable steps, we can ensure accuracy and build a strong understanding of the underlying concepts. The ability to confidently evaluate such expressions is a crucial building block for more advanced mathematical studies. This example highlights the importance of a solid foundation in basic arithmetic and algebraic concepts. The step-by-step approach used here can be applied to a wide range of mathematical problems, emphasizing the importance of clear thinking and careful execution. Mastering these fundamental skills is essential for success in mathematics and related fields.