Evaluating The Expression A Step By Step Solution
In this comprehensive guide, we will meticulously evaluate the expression:
This expression involves a combination of fractions, addition, subtraction, exponentiation, and division. To tackle this, we'll break it down step-by-step, ensuring clarity and accuracy in each operation. Our goal is to express the final answer as a fraction, a whole number, or a mixed number, adhering to the standard conventions of mathematical representation. Before we dive into the solution, let's emphasize the importance of understanding the order of operations (PEMDAS/BODMAS), which dictates the sequence in which we perform mathematical operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This principle will be our guiding light as we navigate through the complexities of the expression. So, let's embark on this mathematical journey together, unraveling the expression piece by piece until we arrive at the final, simplified answer.
Step 1: Simplify the exponent
Our first task is to simplify the exponent within the expression. We encounter the term (-1/2)², which signifies (-1/2) multiplied by itself. Let's delve into the intricacies of exponentiation with fractions. When we square a fraction, we square both the numerator and the denominator. In this case, (-1/2)² translates to (-1)² / (2)². The square of -1 is 1, and the square of 2 is 4. Therefore, (-1/2)² simplifies to 1/4. This step is crucial as it eliminates the exponent, allowing us to proceed with the remaining operations. By addressing the exponent first, we adhere to the order of operations (PEMDAS/BODMAS), ensuring that we're on the right track towards simplifying the expression. Now that we've successfully simplified the exponent, our expression transforms into: (1/5 + -5/8 - 1/4) ÷ 1/4. With the exponent resolved, we're poised to tackle the next phase of simplification, which involves addressing the addition and subtraction within the parentheses. So, let's move forward with confidence, building upon the foundation we've established in this initial step.
Step 2: Find a common denominator for the fractions inside the parentheses
Now, let's tackle the fractions nestled inside the parentheses. We have 1/5, -5/8, and -1/4. To seamlessly add and subtract these fractions, we must first discover a common denominator. This common denominator will serve as the bedrock upon which we can perform the arithmetic operations. To find the common denominator, we seek the least common multiple (LCM) of the denominators 5, 8, and 4. The LCM is the smallest number that is a multiple of all the denominators. In this case, the LCM of 5, 8, and 4 is 40. With the common denominator in hand, we're ready to convert each fraction to an equivalent fraction with a denominator of 40. For 1/5, we multiply both the numerator and denominator by 8, yielding 8/40. For -5/8, we multiply both the numerator and denominator by 5, resulting in -25/40. And for -1/4, we multiply both the numerator and denominator by 10, giving us -10/40. By transforming the fractions to equivalent fractions with a common denominator, we've paved the way for the addition and subtraction operations. Our expression now stands as: (8/40 + -25/40 - 10/40) ÷ 1/4. We're one step closer to simplifying the expression, with the fractions neatly aligned for the next phase of computation.
Step 3: Perform the addition and subtraction inside the parentheses
With the fractions sharing a common denominator, we can now seamlessly perform the addition and subtraction operations within the parentheses. We have 8/40 + -25/40 - 10/40. Let's break it down step-by-step. First, we add 8/40 and -25/40. Adding a negative number is akin to subtraction, so we effectively have 8/40 - 25/40. This yields -17/40. Now, we subtract 10/40 from -17/40. Subtracting a positive number from a negative number results in a more negative number. So, -17/40 - 10/40 equals -27/40. Thus, the expression within the parentheses simplifies to -27/40. This step is pivotal as it consolidates the fractions into a single term, streamlining the expression. Our expression now takes the form: -27/40 ÷ 1/4. We've successfully navigated the addition and subtraction operations, paving the way for the final step: division. With the expression simplified to this stage, we're well-poised to perform the division and arrive at the ultimate answer.
Step 4: Divide by the fraction
Now, we arrive at the final hurdle: dividing -27/40 by 1/4. Dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator. So, the reciprocal of 1/4 is 4/1, which simplifies to 4. Therefore, dividing -27/40 by 1/4 is the same as multiplying -27/40 by 4. Let's perform this multiplication. We have -27/40 multiplied by 4. To multiply a fraction by a whole number, we multiply the numerator by the whole number and keep the denominator the same. So, -27/40 multiplied by 4 equals (-27 * 4) / 40, which simplifies to -108/40. We're now faced with the fraction -108/40. However, this fraction can be simplified further. Both the numerator and denominator share a common factor of 4. Dividing both -108 and 40 by 4, we get -27/10. So, -108/40 simplifies to -27/10. This is an improper fraction, where the numerator is greater than the denominator. To express it as a mixed number, we divide 27 by 10. 27 divided by 10 gives us a quotient of 2 and a remainder of 7. Therefore, -27/10 can be expressed as the mixed number -2 7/10. And there we have it! We've successfully navigated through the expression, performing the division and simplifying the result. The final answer, expressed as a mixed number, is -2 7/10.
Final Answer
Therefore, the final answer, expressed as a mixed number, is -2 7/10. This comprehensive step-by-step solution demonstrates the meticulous process of evaluating the given expression, adhering to the order of operations and simplifying fractions to their final form.
