Solve Numerical Expressions: Step-by-Step Guide

by Marta Kowalska 48 views

Numerical expressions are fundamental in mathematics, combining numbers and operations to form meaningful calculations. Solving these expressions requires a clear understanding of the order of operations and the properties of different mathematical functions. In this article, we will break down several numerical expressions, providing step-by-step solutions and explanations to help you master this essential math skill.

14 + (50 - 18) + 23

Let's start with our first numerical expression: 14 + (50 - 18) + 23. To solve this, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This means we first handle any operations inside parentheses, then exponents, followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right).

Step-by-Step Solution

  1. Parentheses: Inside the parentheses, we have 50 - 18, which equals 32. So, our expression now looks like 14 + 32 + 23.
  2. Addition: Next, we perform the additions from left to right. First, 14 + 32 equals 46. Now we have 46 + 23.
  3. Final Addition: Adding 46 and 23 gives us 69. Therefore, the solution to the numerical expression 14 + (50 - 18) + 23 is 69.

Understanding the Process

The key to solving numerical expressions is to break them down into smaller, manageable steps. By following the order of operations, you can ensure that you arrive at the correct answer. In this case, dealing with the parentheses first simplified the expression, making it easier to handle the subsequent additions. Remember, accuracy in each step is crucial to avoid errors in the final result.

(12 + 28) ÷ (10 + 5) + 17

Our next challenge is the numerical expression (12 + 28) ÷ (10 + 5) + 17. This expression involves both parentheses and division, adding a bit more complexity. But don't worry, guys! We'll tackle it step-by-step, making sure everything is crystal clear.

Step-by-Step Solution

  1. Parentheses: We start by solving the operations within the parentheses.
    • First parentheses: 12 + 28 equals 40.
    • Second parentheses: 10 + 5 equals 15. Our expression now becomes 40 ÷ 15 + 17.
  2. Division: Next, we perform the division. 40 ÷ 15 equals approximately 2.67 (or 8/3 if we want to keep it as a fraction). So, we have 2.67 + 17.
  3. Addition: Finally, we add 2.67 to 17, which gives us 19.67. Therefore, the solution to the numerical expression (12 + 28) ÷ (10 + 5) + 17 is approximately 19.67.

Key Takeaways

In this expression, we encountered multiple sets of parentheses and a division operation. It's essential to handle each set of parentheses independently before moving on to the next operation. Division comes before addition, according to PEMDAS, so we divided before adding. Accurate arithmetic is crucial to avoid calculation errors.

25 + (45 - 18) ÷ 3 + 14

Now let's dive into the numerical expression 25 + (45 - 18) ÷ 3 + 14. This one includes parentheses, subtraction, division, and addition, giving us a good review of the order of operations.

Step-by-Step Solution

  1. Parentheses: We begin with the operation inside the parentheses: 45 - 18 equals 27. Our expression is now 25 + 27 ÷ 3 + 14.
  2. Division: Next, we perform the division: 27 ÷ 3 equals 9. The expression simplifies to 25 + 9 + 14.
  3. Addition: Now we handle the additions from left to right.
    • First addition: 25 + 9 equals 34.
    • Second addition: 34 + 14 equals 48. Therefore, the solution to the numerical expression 25 + (45 - 18) ÷ 3 + 14 is 48.

Mastering the Order of Operations

This problem highlights the importance of following PEMDAS consistently. By prioritizing the operations within parentheses and then division before addition, we arrive at the correct solution. Breaking down the problem into smaller steps makes it easier to manage and reduces the chance of errors.

15 + [5 × (8 - 6 ÷ 2)]

Let's tackle a slightly more complex expression: 15 + [5 × (8 - 6 ÷ 2)]. This expression includes nested parentheses, which means we have parentheses inside brackets. When dealing with nested parentheses, we work from the innermost set outward.

Step-by-Step Solution

  1. Inner Parentheses: Inside the inner parentheses, we have 8 - 6 ÷ 2. Following the order of operations within this set, we first perform the division: 6 ÷ 2 equals 3. So, the expression within the inner parentheses becomes 8 - 3, which equals 5.
  2. Brackets: Now we replace the inner parentheses with its result, giving us 15 + [5 × 5]. Inside the brackets, we have 5 × 5, which equals 25.
  3. Addition: Finally, we add 15 and 25, which gives us 40. Therefore, the solution to the numerical expression 15 + [5 × (8 - 6 ÷ 2)] is 40.

Dealing with Nested Parentheses

Nested parentheses can seem daunting, but the key is to focus on the innermost set first. By systematically working outward, you simplify the expression step-by-step. Always remember the order of operations within each set of parentheses or brackets.

(5 + [28 ÷ 4] + [2 × (5 - 3)])

Our final numerical expression is (5 + [28 ÷ 4] + [2 × (5 - 3)]). This expression involves brackets, division, multiplication, and addition. Let's break it down to see how we can solve it.

Step-by-Step Solution

  1. Inner Parentheses: We start with the inner parentheses (5 - 3), which equals 2. Now the expression looks like (5 + [28 ÷ 4] + [2 × 2]).
  2. Brackets: Next, we solve the operations inside the brackets.
    • First bracket: 28 ÷ 4 equals 7.
    • Second bracket: 2 × 2 equals 4. Our expression becomes (5 + 7 + 4).
  3. Addition: Finally, we add the numbers inside the parentheses: 5 + 7 + 4 equals 16. Therefore, the solution to the numerical expression (5 + [28 ÷ 4] + [2 × (5 - 3)]) is 16.

Final Thoughts on Numerical Expressions

Solving numerical expressions requires a solid understanding of the order of operations and attention to detail. By following PEMDAS, you can systematically simplify even the most complex expressions. Practice is key, guys, so keep working on these types of problems, and you'll become a pro in no time! Remember to break down each problem into smaller steps, and always double-check your work to ensure accuracy. With these tips, you'll be well-equipped to tackle any numerical expression that comes your way.