Problem 38 - Prealgebra

Question

Use the rule for order of operations along with the rules for addition, subtraction, and multiplication to simplify each of the following expressions. $$-8+3[-4-1]$$

Step-by-Step Solution

Answer

The simplified expression is \(-23\).

Step 1: Simplify Inside the Brackets

First, simplify the expression inside the brackets. You have \(-4 - 1\) inside the brackets. Performing the subtraction gives \(-4 - 1 = -5\). Thus, the expression becomes \(-8 + 3[-5]\).

Step 2: Multiply

Next, perform the multiplication outside the brackets. Multiply 3 by the result inside the brackets: \(3 imes -5 = -15\). Now the expression is \(-8 + (-15)\).

Step 3: Addition and Subtraction

Finally, perform the addition of \(-8\) and \(-15\). Since you're adding a negative number, you actually subtract: \(-8 + (-15) = -8 - 15 = -23\). The simplified expression is \(-23\).

Key Concepts

Addition and Subtraction

Addition and subtraction are foundational arithmetic operations that go hand-in-hand. In the order of operations, addition and subtraction are usually addressed last unless they appear within parentheses or brackets. It's important to keep in mind that adding a negative number is the same as subtracting its absolute value. This can sometimes make addition and subtraction a bit tricky when there are negative numbers involved. For instance, in our example, the expression

• -8 + (-15)

can be translated as subtracting 15 from -8, which results in -23. Remember, always resolve any operations enclosed in brackets or parentheses first, if applicable. Also, when moving from left to right through an equation, subtraction can often involve adding a negative. Just keep the order in mind and you'll be set!
To master this, practice various combinations of numbers to see how the rules work in different scenarios.

Multiplication

Multiplication is a key component in simplifying mathematical expressions, particularly when dealing with grouped terms or negative numbers. When multiplying, the order of the numbers doesn't matter (e.g., 3 x 5 is the same as 5 x 3). However, when negative numbers are involved, it tends to become a bit more challenging.

• For instance, multiplying a positive number and a negative number results in a negative product, such as 3 x (-5) = -15.
• Conversely, multiplying two negative numbers yields a positive result.

In expressions like the example we are working with,

• first, simplify the inside of any brackets or parentheses.
• Only after that, proceed to multiplication.

This structured approach helps ensure accuracy. Mastering multiplication with negative numbers builds confidence and skill in tackling more complex problems later on.

Simplifying Expressions

Simplifying expressions is the process of making a mathematical expression more compact and easier to work with. Whether you're working with a simple sum or a complex equation, simplifying expressions is vital. It involves breaking down and combining like terms, following the order of operations. The order of operations can be remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).

• First, handle operations inside parentheses or brackets.
• Next, deal with any exponents.
• Then, move on to multiplication and division in their order of appearance.
• Last, complete addition and subtraction.

Using these steps accurately simplifies expressions efficiently. For example, in our expression, -8 + 3[-4-1], by sticking to the rules, we first simplified inside the bracket to -5, then multiplied to get -15, and finally combined it with -8 to reach -23. Understanding and following these steps precisely leads to the correct solution. Practice simplifies mastering these operations and builds a strong foundation for future mathematical success.