Evaluating Arithmetic Expressions Step-by-Step Solutions

Arithmetic expressions are the backbone of mathematics and programming. Understanding how to evaluate them correctly is crucial for various fields, from basic calculations to complex algorithms. This comprehensive guide will walk you through the process of evaluating several arithmetic expressions, providing a step-by-step approach to ensure accuracy and clarity. Let's dive in and unravel the solutions to these expressions.

1. 8 × (6 ÷ 3 + 5) × (11 - 4)

When evaluating arithmetic expressions, it's crucial to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures that we perform operations in the correct sequence, leading to the accurate result. Let's apply this to our first expression: 8 × (6 ÷ 3 + 5) × (11 - 4).

First, we tackle the parentheses. Inside the first set of parentheses, we have 6 ÷ 3 + 5. According to PEMDAS, division comes before addition, so we divide 6 by 3, which equals 2. Now we have 2 + 5, which sums up to 7. Moving to the second set of parentheses, we have 11 - 4, which equals 7. Our expression now simplifies to 8 × 7 × 7.

Next, we perform the multiplication operations from left to right. 8 × 7 equals 56. Then, we multiply 56 by 7, which gives us 392. Therefore, the final result of the expression 8 × (6 ÷ 3 + 5) × (11 - 4) is 392. This meticulous, step-by-step approach ensures we don't miss any critical steps and arrive at the correct answer. Arithmetic expressions can appear daunting, but breaking them down into manageable parts using the correct order of operations makes them much simpler to solve. Consistent practice and attention to detail are key to mastering these calculations.

2. 6 ÷ 3 + 5 × 7

To accurately solve the arithmetic expression 6 ÷ 3 + 5 × 7, we must adhere to the established order of operations, also known as PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This order ensures we perform each operation in its correct sequence, thus arriving at the right answer. Let's break down the expression step by step.

First, we address the division and multiplication operations. In the expression 6 ÷ 3 + 5 × 7, division and multiplication come before addition. So, we first perform the division: 6 ÷ 3 = 2. Next, we carry out the multiplication: 5 × 7 = 35. Now the expression simplifies to 2 + 35.

With the division and multiplication completed, we move on to the addition operation. We simply add the two numbers: 2 + 35 = 37. Therefore, the final result of the arithmetic expression 6 ÷ 3 + 5 × 7 is 37. This example clearly illustrates the importance of following the order of operations. Without it, we might perform addition before multiplication or division, leading to an incorrect result. Mastery of this order is fundamental in arithmetic and is crucial for more advanced mathematical calculations. Regular practice and a methodical approach will help you confidently solve such expressions.

3. 6 ÷ 8 × 7

In tackling the arithmetic expression 6 ÷ 8 × 7, it's vital to remember the order of operations, often abbreviated as PEMDAS or BODMAS. This acronym reminds us to perform Parentheses/Brackets first, then Exponents/Orders, followed by Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). Applying this principle ensures that we solve expressions in the correct sequence.

For the expression 6 ÷ 8 × 7, we have division and multiplication. According to the order of operations, multiplication and division are performed from left to right. So, we start by dividing 6 by 8. 6 ÷ 8 can be written as a fraction, which is 6/8. This fraction can be simplified to 3/4 or expressed as a decimal, which is 0.75. Now, the expression becomes 0.75 × 7.

Next, we multiply 0.75 by 7. 0.75 × 7 = 5.25. Therefore, the result of the arithmetic expression 6 ÷ 8 × 7 is 5.25. This example highlights the importance of handling operations of the same precedence correctly. When faced with both multiplication and division, or both addition and subtraction, we proceed from left to right. This ensures that we follow the mathematical convention and arrive at the correct answer. Precision and attention to detail are key in such calculations, and consistent practice helps build confidence in solving these expressions.

4. 6 ÷ 56

To accurately evaluate the arithmetic expression 6 ÷ 56, we need to perform a simple division operation. This expression represents 6 divided by 56. Understanding how to carry out this division is fundamental in arithmetic and serves as a building block for more complex calculations. Let’s explore the steps involved in solving this expression.

The expression 6 ÷ 56 can also be written as a fraction: 6/56. To simplify this fraction, we look for common factors in both the numerator (6) and the denominator (56). Both numbers are divisible by 2. Dividing both 6 and 56 by 2, we get a simplified fraction: 3/28. This simplified fraction represents the most basic form of the division.

To express the result as a decimal, we divide 6 by 56 directly. The result is approximately 0.107142857. For practical purposes, this decimal can be rounded to a certain number of decimal places, depending on the required precision. For example, rounding to three decimal places gives us 0.107. Therefore, the result of the arithmetic expression 6 ÷ 56 is approximately 0.107.

This exercise underscores the importance of understanding fractions and decimals, and how they relate to division. Simplifying fractions makes it easier to understand the proportion, while converting to decimals provides a more precise numerical value. Mastery of these basic arithmetic operations is essential for both everyday calculations and more advanced mathematical problem-solving.

5. 9. 5 (ambiguous, please clarify)

The expression 9. 5 is ambiguous and requires clarification. In mathematics, ambiguous expressions can lead to different interpretations and, consequently, different results. To ensure accuracy and clarity, it’s crucial to understand the intended operation or context before attempting to evaluate such expressions. Let’s delve into the potential meanings of 9. 5 and the importance of clear notation in mathematics.

The notation 9. 5 could potentially represent several different mathematical concepts, and without additional context, it's impossible to determine the correct interpretation. One common interpretation might be that 9. 5 is a decimal number, representing nine and five-tenths. In this case, the expression simply represents the numerical value 9.5. However, without context, we cannot definitively rule out other possibilities.

Another possibility is that there might be an implied operation, such as multiplication. For instance, if there is a variable or another number adjacent to 9. 5 without an explicit operator, it might imply multiplication. For example, 9. 5x would mean 9.5 multiplied by x. Similarly, 9. 5(2) would mean 9.5 multiplied by 2. However, without these adjacent elements or explicit operators, this interpretation is speculative.

To avoid ambiguity, it’s essential to use clear and precise mathematical notation. If 9. 5 is intended to be a decimal number, it’s already clear. However, if an operation is intended, it should be explicitly stated. For example, if multiplication is intended, it should be written as 9.5 × ... or 9.5(...). If the intention is to represent a range or interval, it should be written using interval notation, such as [9, 5] or (9, 5). The lack of clarity in the original expression underscores the importance of precise communication in mathematics.

In conclusion, the expression 9. 5 is ambiguous and requires additional context to be correctly evaluated. Mathematical notation should be clear and unambiguous to avoid confusion and ensure accurate results. When faced with ambiguous expressions, it’s always best to seek clarification before proceeding with any calculations.

6. 8 × (36 ÷ 2) - (56 + 20)

To accurately evaluate the arithmetic expression 8 × (36 ÷ 2) - (56 + 20), we must meticulously follow the order of operations, often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This structured approach ensures that we perform each operation in its correct sequence, leading to the accurate result. Let’s break down the expression step by step.

First, we address the operations within the parentheses. In the expression 8 × (36 ÷ 2) - (56 + 20), we have two sets of parentheses. We start with the first set: (36 ÷ 2). Dividing 36 by 2 gives us 18. Now, let's move to the second set of parentheses: (56 + 20). Adding 56 and 20 gives us 76. So, the expression now simplifies to 8 × 18 - 76.

Next, we perform the multiplication operation. We have 8 × 18, which equals 144. The expression now becomes 144 - 76.

Finally, we carry out the subtraction operation. Subtracting 76 from 144 gives us 68. Therefore, the final result of the arithmetic expression 8 × (36 ÷ 2) - (56 + 20) is 68. This detailed, step-by-step evaluation highlights the importance of adhering to the order of operations. Failing to do so can lead to incorrect results. By systematically working through the expression, addressing parentheses first, then multiplication, and finally subtraction, we arrive at the correct solution. Consistent application of PEMDAS is crucial for mastering arithmetic expressions and ensuring accuracy in mathematical calculations.