Problem 61 - Prealgebra

Question

Use the rule for order of operations to simplify each of the following. $$[57+(-35)]+[19+(-24)]$$

Step-by-Step Solution

Answer

The simplified expression is 17.

Step 1: Simplify Inside the Brackets

The expression contains two sets of brackets: \([57 + (-35)]\) and \([19 + (-24)]\). Let's simplify each set individually. First, we simplify \(57 + (-35)\):\[57 - 35 = 22\]So, this expression reduces to 22. Now, simplify \(19 + (-24)\):\[19 - 24 = -5\]This expression reduces to -5.

Step 2: Combine the Results

Now that both bracketed expressions are simplified, the overall expression becomes:\[22 + (-5)\]This can be simplified by performing the addition operation:\[22 - 5 = 17\]

Key Concepts

Simplifying Expressions

When you encounter mathematical expressions involving multiple numbers, symbols, and operations, simplifying them is essential. Simplification helps us understand and solve the problems more efficiently. In our exercise, this involves breaking down the expression into smaller, more manageable parts.

Here, the expression \[[57+(-35)]+[19+(-24)]\] has two main parts enclosed in brackets. Simplifying involves working out each part separately before combining them.

• For \(57 + (-35)\), we rewrite it as \(57 - 35\), because adding a negative number is the same as subtraction.
• Similarly, for \(19 + (-24)\), we convert it to \(19 - 24\).

Simplifying like this makes it easier to handle each section one at a time, leading to clear results: 22 and -5, respectively.

Addition of Integers

Integer addition can sometimes be tricky, especially when dealing with positive and negative numbers. Remember, adding integers is not just about combining numbers; it involves understanding how positive and negative numbers interact.

• When adding a positive number, you move up the number line.
• When adding a negative number, you're actually moving down.

In the example, when we compute \(57 + (-35)\), think of it as how much you "owe" going against how much you "have." Subtract 35 from 57 to get 22.

In the case of \(19 + (-24)\), since 24 is larger than 19, it leads to a negative result, -5. By viewing addition through this lens, problems become simpler to solve.

Mathematical Operations Precedence

Understanding the precedence of mathematical operations is crucial for solving expressions accurately. Precedence tells us the order in which operations should be carried out, ensuring the expression is simplified correctly.

The order of operations is often remembered by the acronym PEMDAS:

• P - Parentheses first
• E - Exponents (ie Powers and Square Roots, etc.)
• M and D - Multiplication and Division (left-to-right)
• A and S - Addition and Subtraction (left-to-right)

In the given expression, \[[57+(-35)]+[19+(-24)]\], parentheses indicate these operations should be completed first.

After simplifying inside the brackets, addition and subtraction are performed according to their precedence. This structured approach ensures that expressions are solved in an orderly and correct manner.