Problem 46 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ -8(4-12)+2 $$

Step-by-Step Solution

Answer

The value is 66.

Step 1: Simplify the Expression Inside the Parentheses

Start by simplifying the expression inside the parentheses: \(4 - 12\). Calculate the difference.\[4 - 12 = -8\]

Step 2: Multiply by the Coefficient

After simplifying the parentheses, multiply the result by \(-8\). Use the simplified expression \(-8\) from Step 1.\[-8 \times (-8) = 64\]

Step 3: Add the Remaining Number

Now that you've multiplied, add the remaining number, \(+2\), to the result from Step 2.\[64 + 2 = 66\]

Step 4: Verify Using a Calculator

Double-check your computations by using a calculator. Enter \(-8(4 - 12) + 2\) and ensure that the result matches \(66\).

Key Concepts

Parentheses Simplification

When faced with an algebraic expression that includes parentheses, it's crucial to simplify what's inside them as the first step. The expression given was \(4 - 12\). In mathematics, parentheses act as signals telling you to prioritize the calculations within. Here’s how you approach it:

• Start by looking at the numbers and operations in the parentheses.
• Perform the operations inside parentheses first before any other operation.
• In our case: \(4 - 12\):
• Subtract 12 from 4 which equals \(-8\).

Understanding how to handle computations within parentheses ensures that we're setting up the rest of the equation correctly. It's like laying a solid foundation for a good start in solving any math problem.

Multiplication of Integers

After simplifying expressions within the parentheses, the next step in the problem was multiplying integers. Multiplication is often represented by placing a number immediately next to parentheses, indicating that all numbers within the parentheses should be multiplied by it.

We had: \(-8 imes (-8)\). Here's the breakdown:

• Notice that both numbers involved are negative. This is important.
• In math, multiplying two negative integers results in a positive product.
• So, \(-8 imes (-8) = 64\).

Understanding integer rules can greatly simplify solving algebraic equations. In this instance, knowing that a negative times a negative equals a positive helps us maintain accuracy.

Addition of Integers

In the final step of solving the equation, addition of integers takes the spotlight. Addition is the process of combining numbers to get a total sum. Let's break down what happened:

After multiplying your integers, you ended with \(64\). The next number that needs to be added is \(+2\). Here's how to think about this:

• Start by looking at your current result: \(64\).
• Add the integer \(+2\) to \(64\). Think of this as moving two units to the right on a number line.
• The result is \(66\), which is the final sum.

Remember that being comfortable with adding integers will make it much easier to work through math problems efficiently. The key is to be mindful of the integers and always double-check your calculations if necessary.