Solving (1/2)^3 - 4 ÷ √64 A Step By Step Guide
This article delves into the step-by-step solution of the mathematical expression (1/2)^3 - 4 ÷ √64. We will break down each component, ensuring a clear and comprehensive understanding of the process involved. This expression combines several fundamental mathematical operations, including exponentiation, division, and the extraction of square roots. By carefully examining each step, we aim to provide a robust guide for anyone looking to enhance their mathematical problem-solving skills. Understanding the order of operations, often remembered by the acronym PEMDAS/BODMAS, is crucial for correctly solving such expressions. We will explore how this order guides us through the calculations, ensuring accuracy and clarity in our solution. Furthermore, we'll emphasize the importance of mastering these basic operations as they form the foundation for more complex mathematical concepts. Let's begin by dissecting the expression and addressing each operation systematically.
Breaking Down the Expression
The expression (1/2)^3 - 4 ÷ √64 can be intimidating at first glance, but by breaking it down into smaller, manageable parts, we can simplify the process significantly. The first part we encounter is (1/2)^3, which involves exponentiation. This means we need to raise the fraction 1/2 to the power of 3. This operation is crucial in various mathematical fields, including algebra and calculus, and mastering it is essential for overall mathematical proficiency. Next, we have the division operation 4 ÷ √64. This requires us to first find the square root of 64 and then divide 4 by the result. Understanding square roots is fundamental in mathematics, as they appear in numerous contexts, from geometry to number theory. Lastly, we have the subtraction operation connecting the two main parts of the expression. Subtraction, as a basic arithmetic operation, is something we encounter daily, and its correct application is vital for accuracy in calculations. By addressing each of these components individually and then combining the results, we can effectively solve the entire expression. The following sections will provide a detailed step-by-step solution, ensuring a clear understanding of each operation involved.
Step 1: Evaluating (1/2)^3
In this first step, we focus on evaluating the exponential term (1/2)^3. Exponentiation involves raising a base to a certain power, which in this case means multiplying the base (1/2) by itself the number of times indicated by the exponent (3). So, (1/2)^3 is equivalent to (1/2) * (1/2) * (1/2). When multiplying fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. Thus, (1/2) * (1/2) * (1/2) = (111) / (222) = 1/8. Understanding exponentiation is crucial as it appears in various mathematical contexts, including polynomial equations, exponential growth, and calculus. Moreover, proficiency in manipulating exponents is fundamental for simplifying expressions and solving equations. This initial step demonstrates the importance of accurately applying the rules of exponents to arrive at the correct result. Once we have evaluated (1/2)^3 to be 1/8, we can move on to the next operation in the expression. This step-by-step approach allows us to manage complex expressions by breaking them down into smaller, more manageable components, ensuring a higher level of accuracy and understanding. Let's now proceed to the next step in solving the expression.
Step 2: Calculating the Square Root of 64
The next step in solving the expression (1/2)^3 - 4 ÷ √64 involves calculating the square root of 64, denoted as √64. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we are looking for a number that, when multiplied by itself, equals 64. By recalling our knowledge of multiplication tables or using a calculator, we can determine that 8 * 8 = 64. Therefore, the square root of 64 is 8. Understanding square roots is fundamental in mathematics, particularly in geometry when dealing with areas and lengths, and in algebra when solving quadratic equations. The ability to quickly identify common square roots can significantly speed up problem-solving. Now that we have found √64 to be 8, we can proceed with the division operation in the expression. This step emphasizes the importance of knowing basic mathematical facts and operations, as they form the building blocks for more complex calculations. With the square root determined, we can now move on to the next step, which involves dividing 4 by the square root we just calculated.
Step 3: Performing the Division 4 ÷ √64
Having calculated √64 to be 8 in the previous step, we now proceed with the division operation 4 ÷ √64, which translates to 4 ÷ 8. Division is a fundamental arithmetic operation that involves splitting a quantity into equal parts. In this case, we are dividing 4 by 8. This can be expressed as the fraction 4/8. To simplify this fraction, we can find the greatest common divisor (GCD) of both the numerator (4) and the denominator (8). The GCD of 4 and 8 is 4. Dividing both the numerator and the denominator by 4, we get 4/4 = 1 and 8/4 = 2. Therefore, the simplified fraction is 1/2. Alternatively, one can recognize that 4 is half of 8, hence 4 ÷ 8 equals 1/2 or 0.5. This division operation highlights the importance of simplifying fractions to arrive at the most concise and understandable form of the result. Proficiency in division and fraction simplification is crucial for accurate mathematical calculations. With the division operation completed, we now have the result 1/2. This allows us to move on to the final step of the expression, which involves subtracting this result from the value obtained in Step 1. Let’s proceed to the final step to complete the solution.
Step 4: Completing the Subtraction (1/8) - (1/2)
In the final step, we need to perform the subtraction (1/8) - (1/2). To subtract fractions, they must have a common denominator. The smallest common denominator for 8 and 2 is 8. We need to convert the fraction 1/2 into an equivalent fraction with a denominator of 8. To do this, we multiply both the numerator and the denominator of 1/2 by 4: (1 * 4) / (2 * 4) = 4/8. Now, our subtraction becomes (1/8) - (4/8). When subtracting fractions with a common denominator, we subtract the numerators while keeping the denominator the same. So, 1/8 - 4/8 = (1 - 4) / 8 = -3/8. The result is a negative fraction, which indicates that the second term (1/2) is larger than the first term (1/8). This subtraction step showcases the importance of understanding fraction arithmetic, including finding common denominators and performing subtraction with both positive and negative fractions. This skill is fundamental for solving more complex algebraic equations and mathematical problems. Therefore, the final result of the expression (1/2)^3 - 4 ÷ √64 is -3/8. This comprehensive step-by-step solution ensures a clear understanding of each operation and the overall process.
Final Answer: -3/8
In conclusion, the step-by-step solution of the mathematical expression (1/2)^3 - 4 ÷ √64 has led us to the final answer of -3/8. This process involved several key mathematical operations, including exponentiation, square root extraction, division, and subtraction. We started by evaluating (1/2)^3, which resulted in 1/8. Next, we calculated the square root of 64, finding it to be 8. We then performed the division 4 ÷ 8, which simplified to 1/2. Finally, we completed the subtraction (1/8) - (1/2), which, after finding a common denominator and performing the subtraction, yielded the result -3/8. This solution highlights the importance of adhering to the order of operations (PEMDAS/BODMAS) and the necessity of understanding fundamental arithmetic principles. Each step was carefully dissected to provide clarity and ensure a comprehensive understanding of the process. Mastering such problem-solving techniques is crucial for success in mathematics and related fields. The ability to break down complex expressions into manageable steps is a valuable skill that enhances both accuracy and confidence in mathematical endeavors. We hope this detailed explanation has provided a clear and helpful guide for solving similar mathematical expressions.
