Problem 30 - 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] $$\frac{-4(-12)}{-6}$$

Step-by-Step Solution

Answer

The expression simplifies to \(-8\).

Step 1: Simplify the Expression in the Numerator

Look at the expression in the numerator, \(-4 \times (-12)\). According to the rules of multiplication, the product of a negative number and another negative number is positive. So, multiply \(-4\) by \(-12\).Calculation: \(-4 \times (-12) = 48.\)

Step 2: Divide the Result by the Denominator

Now divide the simplified numerator by the denominator. We have:\(\frac{48}{-6}\).According to the rules of division, dividing a positive number by a negative number results in a negative number. So:Calculation: \(48 \div (-6) = -8.\)

Key Concepts

Multiplication Rules

Understanding multiplication rules is crucial for solving problems involving multiple numbers. These rules help determine the sign and value of your final product.

- Multiplying two positive numbers always gives a positive product. For example, \(3 \times 4 = 12\).- Likewise, multiplying two negative numbers also yields a positive result. This is because the negatives cancel each other out. For instance, \(-3 \times -4 = 12\), just like they were both positive.- When multiplying a positive and a negative number, the product will be negative. For example, \(3 \times -4 = -12\).
Memorize these sign rules to easily find the product of different numbers. Remember, the sign depends on whether the numbers have the same or different signs.

Division Rules

Division rules are just as important as multiplication rules when simplifying expressions. They help decide the sign and result of division operations.

- If you divide two numbers with the same sign (both positive or both negative), the quotient is positive. For example, \(\frac{6}{3} = 2\) and \(\frac{-6}{-3} = 2\).- However, dividing numbers with different signs yields a negative quotient. So, \(\frac{6}{-3} = -2\) and \(\frac{-6}{3} = -2\).
Always keep an eye on the signs before performing division. It's essential to apply them correctly to get the right result. Similar to multiplication, the signs determine whether your answer will be positive or negative.

Simplifying Expressions

The process of simplifying expressions involves applying basic arithmetic operations and rules to make calculations easier and clearer.

When you approach an expression:

• Start by looking at the terms involved and identify any operations that need priority based on the order of operations (PEMDAS/BODMAS).
• First handle any calculations within parentheses or brackets.
• Next, look for any exponents.
• Then proceed with multiplication and division from left to right.
• Finally, handle any addition and subtraction from left to right.

In the example \(\frac{-4(-12)}{-6}\), simplifying involves multiplying the two negatives in the numerator first to get 48, and then dividing by -6 to reach the final simplified answer of -8. Ensuring you practice each part in turn helps avoid mistakes and leads to accurate results.