Problem 45 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ -3(1-6) $$

Step-by-Step Solution

Answer

The value is 15.

Step 1: Simplify Inside the Parentheses

First, simplify the expression inside the parentheses by subtracting 6 from 1. This can be done by performing the operation: \[1 - 6 = -5\].

Step 2: Multiply the Result

Now, multiply the number you obtained from Step 1 with -3. This will give the final value of the expression:\[-3 \times (-5) = 15\].

Step 3: Check with a Calculator

Use a calculator to double-check your result. Enter the operation exactly as it appears: 1. Enter `1 - 6` which yields `-5`. 2. Multiply the result by `-3`, which should result in `15`.

Key Concepts

Parentheses in Math

Parentheses are an essential part of mathematical expressions. They help us group numbers and operations together, ensuring calculations occur in the correct order. This is why phrases like "please excuse my dear Aunt Sally" help students remember the Order of Operations (PEMDAS/BODMAS). Parentheses are the first priority when solving equations.

They signal which operations to perform first, regardless of the general order of operations.

In this exercise, the parentheses around \(1 - 6\) indicate that you must perform the subtraction before doing anything else.

Simplifying inside the parentheses first ensures accurate calculations and prevents mistakes. Always do what is inside parentheses immediately, before moving on to multiplication, division, or other operations.
Using parentheses correctly can change the meaning of an expression completely, so it's crucial to use them appropriately.

Negative Numbers Multiplication

Multiplying negative numbers can sometimes seem confusing, but there's a simple rule to follow. When you multiply a negative number by another negative number, the result is a positive number.

Think of negative signs as indicators of opposites or flips.

When two negatives interact, they "flip" to form a positive.

In our example, \(-3\) is multiplied by \(-5\). Following the rule of multiplying negatives gives us a positive 15: \(-3 \times -5 = 15\).
Remember, the outcome is positive because switching direction (from negative to positive) twice brings you back to facing forward.

Simplifying Expressions

Simplifying expressions is like cleaning up a mess—organizing and resolving it till nothing more can be simplified. You break down complicated expressions into simpler equations that are much easier to solve.
In this exercise, the expression begins with \(-3(1-6)\).

Firstly, simplify inside the parentheses. This leads to a single value \(-5\).

Next, multiply \(-3\) by this result.

Each step is a form of simplification, refining the expression by progressively removing unnecessary complexity.
With practice, simplifying expressions becomes an automatic part of solving math problems—making them less daunting and more manageable.