- Identify the order of operations: prioritize exponents, then multiplication, then addition.
- Evaluate the exponent part: \(\left(-\frac{1}{4}\right)^2 = \frac{1}{16}\).
- Calculate the product: \(-\frac{1}{2} \times \frac{3}{4} = -\frac{3}{8}\).
- Find a common denominator to add the terms: \(-\frac{6}{16} + \frac{1}{16} = \boxed{-\frac{5}{16}}\).
Explanation
- Problem Analysis and Order of Operations
We are given the expression \[-\frac{1}{2} \times \frac{3}{4} + \left(-\frac{1}{4}\right)^2.\] To evaluate this, we must strictly follow the order of operations (often referred to as PEMDAS or BODMAS), which dictates the sequence: parentheses, exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right). Our plan is to evaluate the exponent, perform the multiplication, and then add the two resulting terms.
- Calculating the Exponent and Product
First, we address the exponentiation term: \[\left(-\frac{1}{4}\right)^2.\] Since the square of a negative number is positive, this becomes \[\left(-\frac{1}{4}\right) \times \left(-\frac{1}{4}\right) = \frac{1}{16}.\] Next, we perform the multiplication term: \[-\frac{1}{2} \times \frac{3}{4} = -\frac{1 \times 3}{2 \times 4} = -\frac{3}{8}.\] We now substitute these back into the original expression.
- Final Addition and Simplification
Now we combine the results: \[-\frac{3}{8} + \frac{1}{16}.\] To add these two fractions, we need a common denominator. The least common multiple of 8 and 16 is 16. We convert the first fraction by multiplying the numerator and denominator by 2: \[-\frac{3}{8} = -\frac{3 \times 2}{8 \times 2} = -\frac{6}{16}.\] Adding them together: \[-\frac{6}{16} + \frac{1}{16} = \frac{-6 + 1}{16} = -\frac{5}{16}.\] Thus, the final value is \(-\frac{5}{16}\).
Examples
Order of operations is crucial in everyday life, especially in fields like finance and engineering. For instance, when calculating the total cost of a project where parts are ordered in batches and there is a discount or fixed overhead, it's essential to compute the exponentiation (like interest rates over time) and multiplication before adding the total. Understanding these rules ensures that you don't overspend or miscalculate budgets, just as we did here to arrive at the precise value of \(-\frac{5}{16}\).