Evaluating The Expression 5 * 3 - (7 + 3)^2 A Step-by-Step Guide

In this article, we will delve into the step-by-step evaluation of the mathematical expression $5 \cdot 3 - (7+3)^2$. This expression involves a combination of arithmetic operations, including multiplication, addition, and exponentiation. Mastering the order of operations, often remembered by the acronym PEMDAS/BODMAS, is crucial for accurately simplifying such expressions. This acronym dictates the sequence in which operations should be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). By carefully following these rules, we can break down the expression into manageable steps and arrive at the correct solution. This detailed exploration will not only provide the final answer but also enhance your understanding of fundamental mathematical principles. The ability to evaluate expressions accurately is a foundational skill in mathematics, essential for tackling more complex problems in algebra, calculus, and beyond. Throughout this article, we will emphasize the importance of each step, ensuring clarity and fostering a strong grasp of the underlying concepts.

To accurately evaluate the expression $5 \cdot 3 - (7+3)^2$, we must adhere to the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This systematic approach ensures that we perform the operations in the correct sequence, leading to the accurate simplification of the expression.

1. Parentheses

The first step in evaluating our expression is to address the parentheses. Inside the parentheses, we have the addition operation $7 + 3$. Performing this addition yields:

$7 + 3 = 10$

So, the expression now becomes:

$5 \cdot 3 - (10)^2$

By simplifying the expression within the parentheses, we've taken the first critical step in adhering to the order of operations. This simplification is essential because it prepares us for the next operation, which is exponentiation.

2. Exponents

After addressing the parentheses, the next operation to consider according to PEMDAS is exponentiation. In our simplified expression, $5 \cdot 3 - (10)^2$, we have the term $(10)^2$, which means 10 raised to the power of 2. This operation is equivalent to multiplying 10 by itself:

$(10)^2 = 10 \cdot 10 = 100$

Now, substituting this result back into our expression, we get:

$5 \cdot 3 - 100$

The exponentiation step has effectively reduced the complexity of the expression, setting the stage for the multiplication and subtraction operations that follow. It's crucial to handle exponents before other operations to maintain the correct order and arrive at the accurate final result.

3. Multiplication

With the parentheses and exponents taken care of, the next operation in the order of operations is multiplication. In our expression, $5 \cdot 3 - 100$, we have a multiplication operation: $5 \cdot 3$. Performing this multiplication gives us:

$5 \cdot 3 = 15$

Substituting this result back into our expression, we now have:

$15 - 100$

This step highlights the importance of adhering to the order of operations, as performing the multiplication before the subtraction is crucial for arriving at the correct answer. By systematically addressing each operation in the appropriate sequence, we ensure the accuracy of our calculations.

4. Subtraction

The final operation to perform in our expression, $15 - 100$, is subtraction. Subtracting 100 from 15 yields:

$15 - 100 = -85$

Therefore, the final result of evaluating the expression $5 \cdot 3 - (7+3)^2$ is -85. This step concludes our step-by-step evaluation, demonstrating how following the order of operations leads us to the correct solution. Subtraction, being the last operation in this particular expression, ties together all the previous steps, resulting in the final numerical answer.

In summary, by meticulously following the order of operations (PEMDAS), we have successfully evaluated the expression $5 \cdot 3 - (7+3)^2$. We began by simplifying the expression within the parentheses, followed by addressing the exponent, then performing the multiplication, and finally, carrying out the subtraction. Each step was crucial in arriving at the correct answer. The final result of the evaluation is:

$5 \cdot 3 - (7+3)^2 = -85$

This comprehensive evaluation underscores the significance of adhering to mathematical conventions and the order of operations. A clear understanding of these principles is essential for accurate problem-solving in mathematics and related fields. The process of breaking down a complex expression into simpler, manageable steps not only aids in finding the correct solution but also enhances one's mathematical reasoning and analytical skills. This skill is invaluable for tackling more complex mathematical challenges in the future.

In conclusion, we have successfully evaluated the expression $5 \cdot 3 - (7+3)^2$ by meticulously following the order of operations, often remembered by the acronym PEMDAS/BODMAS. This systematic approach involved simplifying the parentheses first, then addressing the exponent, followed by multiplication, and finally, performing the subtraction. Each step was critical in arriving at the correct final answer, which is -85. This exercise highlights the fundamental importance of adhering to mathematical conventions to ensure accuracy in calculations. Understanding and applying the order of operations is not just a mathematical skill but a crucial tool for problem-solving in various disciplines. The ability to break down complex problems into smaller, manageable steps is a valuable asset that extends beyond the realm of mathematics. By mastering these basic principles, we lay a strong foundation for tackling more advanced mathematical concepts and real-world applications. This comprehensive evaluation serves as a testament to the power of structured thinking and the importance of precision in mathematical computations.