Problem 44 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ 8(5-12) $$

Step-by-Step Solution

Answer

The value of the expression is \(-56\).

Step 1: Parentheses First

In any mathematical expression, operations inside parentheses get priority. Here, you have the expression \(5 - 12\) inside the parentheses. Calculate it first: \(5 - 12 = -7\).

Step 2: Multiply

Now, take the result from the parentheses and proceed to perform the multiplication. Multiply \(8\) by \(-7\), yielding \(8 imes (-7) = -56\).

Step 3: Verify with Calculator

Use a calculator to recheck your computations. Input the expression \(8 imes (5 - 12)\). The calculator should display \(-56\), confirming our manual calculations.

Key Concepts

Parentheses

When solving mathematical expressions, it's crucial to start with operations inside parentheses. This is because parentheses indicate that what's inside them must be calculated first, which is a fundamental part of the order of operations—often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). In our example expression, we have

• \(5 - 12\)

located within parentheses. We calculate this first, resulting in

• \(5 - 12 = -7\)

.
Ignoring the parentheses might lead to incorrect results because subsequent operations might be performed in the wrong order. Ensure all expressions within parentheses are simplified before moving on to other operations.

Integer Operations

After solving expressions within parentheses, the next step is performing integer operations, such as multiplication, division, addition, or subtraction. In our case, the expression reduces to a multiplication after handling the parentheses.
Multiply the result from inside the parentheses by any numbers outside it. Here, you need to multiply eight by the negative seven we obtained:

• \(8 \times (-7)\)

.
This operation involves multiplication with a negative integer. The rule for multiplying integers states:

• Multiplying two positive integers or two negative integers gives a positive result.
• Multiplying a positive integer by a negative integer gives a negative result.

Applying this, \(8 \times (-7)\) yields

• \(-56\)

.

Calculator Verification

Verification with a calculator is a practical step to ensure accuracy in math. This is especially helpful for confirming steps in more complex calculations or when number errors could significantly affect results.
To verify our example, input the expression as it appears:

• \(8 \times (5 - 12)\)

on your calculator.
Most calculators follow the correct order of operations automatically, so they will handle the parentheses first, calculating \(5 - 12\) as \(-7\) and then multiplying by eight to give \(-56\).
Ensure your calculator is set to the correct mode (if applicable) and check that each entry is correct before pressing equals. This practice is not just about confirming the correctness of your answer; it also helps cultivate careful habits when performing calculations.