Problem 41 - Prealgebra

Question

Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the following expressions as much as possible. [Examples 6–9] $$\frac{3(-7)(-4)}{6(-2)}$$

Step-by-Step Solution

Answer

The simplified expression is \(-7\).

Step 1: Simplify the Numerator

First, simplify the numerator of the fraction by multiplying the terms: \(3 \times (-7) \times (-4)\). Start by multiplying \((-7)\) and \((-4)\), which gives \(28\). Then multiply this result by \(3\) to get \(84\). Thus, the simplified numerator is \(84\).

Step 2: Simplify the Denominator

Now, simplify the denominator by multiplying the terms: \(6 \times (-2)\). Multiply these two numbers to get \(-12\). Thus, the simplified denominator is \(-12\).

Step 3: Divide Numerator by Denominator

Divide the simplified numerator by the simplified denominator: \(\frac{84}{-12}\). Dividing \(84\) by \(-12\) gives \(-7\).

Step 4: Final Simplification and Verification

Ensure there are no further simplifications possible. The result \(-7\) is fully simplified, so no additional steps are necessary.

Key Concepts

Simplifying Fractions

Simplifying fractions means reducing a fraction to its simplest form. This involves making the numerator and denominator as small as possible, while still retaining the fraction's original value. For example, when you have a fraction like \(\frac{84}{-12}\), you find the greatest common divisor (GCD) of the numerator and the denominator to simplify it.
The GCD is the largest number that divides both the numerator and the denominator exactly. In our example, the GCD of 84 and 12 is 12. You divide both 84 and -12 by this number:

• 84 divided by 12 is 7
• -12 divided by 12 is -1

After simplification, the fraction becomes \(\frac{7}{-1}\), which is equal to -7.
This process helps in solving problems faster and makes calculations easier.

Negative Numbers

Managing negative numbers can be tricky at first, but with practice, it becomes second nature. Negative numbers are numbers with a minus sign in front, indicating they are below zero on the number line. When two negative numbers are multiplied together, the result is positive. This rule is crucial when simplifying fractions or solving equations. For instance, multiplying \((-7)\) by \((-4)\) gives you a positive 28. On the contrary, when you multiply a negative number by a positive one, the result is negative. For example \(6 \times (-2) = -12\).
To sum up, remember these two key rules:

• A negative times a negative is a positive.
• A positive times a negative is a negative.

This helps significantly in guiding how you simplify fractions or perform other operations involving negative numbers.

Multiplication Rules

The rules of multiplication are fundamental, especially when dealing with operations involving fractions and negative numbers. Always remember the order of operations, which often involves several steps to solve an expression accurately.
When you look at an expression like \(3 \times (-7) \times (-4)\), start by:

• Multiplying the first negative with the second, \((-7) \times (-4) = 28\).
• Then, multiply this result by the positive number, \(3 \times 28 = 84\).

Applying these rules ensures you keep track of sign changes and perform accurate calculations. The multiplication rule's importance cannot be understated, as incorrect sign management often leads to errors in simplifying and solving problems.