Solving The Mathematical Expression [5^2+(5-7)^2-35 ÷(-7)]+6 A Step-by-Step Guide

Jul 10, 2025 by ADMIN 82 views

Understanding and Solving the Mathematical Expression [5^2+(5-7)^2-35 ÷(-7)]+6

In the realm of mathematics, the ability to solve complex expressions is a fundamental skill. This article delves into the step-by-step solution of the mathematical expression [5^2+(5-7)2-35 ÷(-7)]+6. We will break down each component of the equation, applying the order of operations (PEMDAS/BODMAS) to arrive at the correct answer. This exercise not only enhances our understanding of arithmetic operations but also highlights the importance of precision and methodical problem-solving in mathematical calculations. Whether you are a student looking to improve your algebra skills or someone who enjoys the challenge of numbers, this detailed explanation will provide a comprehensive guide to tackling similar problems. Mastering such expressions builds a strong foundation for more advanced mathematical concepts and real-world applications.

The expression [5^2+(5-7)2-35 ÷(-7)]+6 might seem daunting at first, but by dissecting it into smaller, manageable parts, we can easily solve it. The key to success lies in understanding and applying the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). This order ensures that we perform the operations in the correct sequence, leading to the accurate result. Our initial step involves tackling the operations within the brackets. We start by addressing the exponent, which simplifies to a numerical value. Next, we perform subtraction and division within the brackets, respecting the rules for negative numbers. Once the expressions inside the brackets are simplified, we can proceed to the final addition operation outside the brackets. This systematic approach not only makes the problem more approachable but also reduces the chances of making errors. In the following sections, we will meticulously walk through each of these steps, clarifying the logic behind each operation and illustrating how the final answer is derived. By the end of this explanation, you will have a clear understanding of how to solve similar mathematical problems and appreciate the elegance of structured arithmetic.

To solve the expression [5^2+(5-7)2-35 ÷(-7)]+6, we follow the order of operations (PEMDAS/BODMAS), which dictates the sequence in which mathematical operations should be performed. This systematic approach ensures accuracy and clarity in our calculations. First, we address the operations within the parentheses and brackets, then exponents, followed by multiplication and division, and finally, addition and subtraction. This method helps us break down the complex expression into simpler, more manageable steps. Let’s delve into each stage of the solution.

1. Solve the Innermost Parentheses (5-7)

The first step in simplifying our expression is to address the innermost parentheses: (5-7). This is a straightforward subtraction operation involving integers. Subtracting 7 from 5 gives us a negative result. The calculation is as follows:

(5 - 7) = -2

This result will be crucial as we move forward in solving the expression. We've now simplified one part of the equation, making it easier to manage the rest of the operations. Understanding how to handle integer subtraction is fundamental in mathematics, and this step lays the groundwork for the subsequent calculations. With this initial simplification, we are one step closer to finding the final answer.

2. Calculate the Exponents: 5^2 and (-2)^2

After dealing with the innermost parentheses, the next step according to the order of operations (PEMDAS/BODMAS) is to calculate the exponents. In our expression, we have two exponential terms: 5^2 and (-2)^2. Let's calculate each of these separately to avoid confusion. Exponents indicate repeated multiplication, so we will multiply each base number by itself.

5^2 means 5 multiplied by itself, which is 5 * 5. 5^2 = 5 * 5 = 25
(-2)^2 means -2 multiplied by itself, which is (-2) * (-2). (-2)^2 = (-2) * (-2) = 4

It's important to remember that multiplying two negative numbers results in a positive number. So, (-2) * (-2) equals 4, not -4. With the exponents calculated, we've further simplified our expression. These results will be used in the next steps as we continue to follow the order of operations. Calculating exponents correctly is a vital skill in mathematics, and these steps demonstrate how to apply this concept within a larger expression.

3. Perform the Division: 35 ÷ (-7)

With the exponents evaluated, our next step in solving the expression [5^2+(5-7)2-35 ÷(-7)]+6 is to perform the division operation. According to the order of operations (PEMDAS/BODMAS), division and multiplication should be done before addition and subtraction. In this case, we have 35 ÷ (-7). This is a division problem involving a positive number (35) and a negative number (-7). When dividing a positive number by a negative number, the result will be negative. Let's perform the division:

35 ÷ (-7) = -5

The result is -5. It's crucial to remember the rules of signs in mathematics when performing division. A positive divided by a negative yields a negative quotient. With this division completed, we have further simplified our expression, and we are now closer to the final answer. The ability to correctly perform division, especially with negative numbers, is a fundamental skill in solving mathematical problems.

4. Substitute the Calculated Values into the Expression

Now that we have calculated the individual components of the expression [5^2+(5-7)2-35 ÷(-7)]+6, it’s time to substitute these values back into the original expression. This step allows us to consolidate our calculations and move towards the final solution. We have found that:

5^2 = 25
(5-7) = -2, and (-2)^2 = 4
35 ÷ (-7) = -5

Substituting these values into the original expression, we get:

[25 + 4 - (-5)] + 6

This substitution simplifies the expression significantly, making it easier to manage. We've replaced the exponents, parentheses, and division with their numerical equivalents. The next step involves simplifying the expression within the brackets, which will bring us closer to the final answer. This methodical approach, of breaking down the problem into smaller parts and then reassembling it, is a key strategy in mathematical problem-solving. It reduces the complexity and makes the process more transparent and less prone to errors.

5. Simplify Inside the Brackets: [25 + 4 - (-5)]

Having substituted the calculated values into our expression, we now focus on simplifying the expression within the brackets: [25 + 4 - (-5)]. According to the order of operations (PEMDAS/BODMAS), we perform addition and subtraction from left to right. It's important to handle the subtraction of a negative number carefully, as subtracting a negative is equivalent to adding a positive.

First, we perform the addition:

25 + 4 = 29

Now our expression within the brackets looks like this:

[29 - (-5)]

Next, we handle the subtraction of the negative number. Subtracting -5 is the same as adding 5:

29 - (-5) = 29 + 5 = 34

So, the expression within the brackets simplifies to 34. With this simplification, our original expression has become much more manageable. We have systematically reduced the complexity by addressing each operation in the correct order. This step-by-step approach is crucial in mathematics to ensure accuracy and clarity. Now we are just one step away from the final solution.

6. Final Addition: 34 + 6

After simplifying the expression inside the brackets to 34, our final step is to perform the remaining addition: 34 + 6. This is a straightforward addition operation that will give us the final answer to the original expression [5^2+(5-7)2-35 ÷(-7)]+6. Let’s perform the addition:

34 + 6 = 40

Therefore, the final result of the expression is 40. We have successfully solved the complex mathematical expression by breaking it down into manageable steps and following the order of operations (PEMDAS/BODMAS). This process illustrates the importance of methodical problem-solving in mathematics, where accuracy and attention to detail are paramount. With this final calculation, we have completed our journey through the expression, arriving at the definitive answer.

In conclusion, the solution to the mathematical expression [5^2+(5-7)2-35 ÷(-7)]+6 is 40. This problem provided an excellent opportunity to reinforce our understanding of the order of operations (PEMDAS/BODMAS) and the rules of arithmetic. By systematically breaking down the expression into smaller parts, we were able to tackle each operation with precision and confidence. Starting with the innermost parentheses, we worked our way through exponents, division, subtraction, and finally, addition. This step-by-step approach not only ensures accuracy but also enhances our problem-solving skills in mathematics. The importance of understanding and applying the correct order of operations cannot be overstated, as it is fundamental to solving a wide range of mathematical problems. Whether you are a student learning algebra or someone who enjoys the mental exercise of numbers, mastering these skills is essential. We hope this detailed explanation has been helpful in clarifying the process and boosting your confidence in tackling similar challenges. Keep practicing and exploring the fascinating world of mathematics!