Problem 61 - Elementary Algebra

Question

Use the commutative properties of addition and multiplication to write equivalent expressions for the following problems. $$ (6)(-9)(-2) $$

Step-by-Step Solution

Answer

Answer: We can rearrange the numbers in the expression (6)(-9)(-2) using the commutative property of multiplication in six different ways: (-9)(6)(-2), (-2)(6)(-9), (-2)(-9)(6), (-9)(-2)(6), and (6)(-2)(-9). All the equivalent expressions result in the same value, 108.

Step 1: Understand the commutative property of multiplication

The commutative property of multiplication states that for any numbers a, b, and c, a * b * c is the same as c * b * a, or any other order of the numbers, as long as the operation between them is multiplication. In this problem, we have the numbers 6, -9, and -2. We will use the commutative property to rearrange these numbers in different ways, showing that the result will be the same.

Step 2: Rearrange the numbers

First, let's rearrange the numbers and find the equivalent expressions: 1. Original expression: \((6)(-9)(-2)\) 2. Rearrange #1: \((-9)(6)(-2)\) 3. Rearrange #2: \((-2)(6)(-9)\) 4. Rearrange #3: \((-2)(-9)(6)\) 5. Rearrange #4: \((-9)(-2)(6)\) 6. Rearrange #5: \((6)(-2)(-9)\) These are the six possible combinations when rearranging the integers using the commutative property of multiplication.

Step 3: Calculate the results

Now, let's calculate the result for each equivalent expression: 1. Original expression: \((6)(-9)(-2) = -54 * -2 = 108\) 2. Rearrange #1: \((-9)(6)(-2) = -54 * -2 = 108\) 3. Rearrange #2: \((-2)(6)(-9) = -12 * -9 = 108\) 4. Rearrange #3: \((-2)(-9)(6) = 18 * 6 = 108\) 5. Rearrange #4: \((-9)(-2)(6) = 18 * 6 = 108\) 6. Rearrange #5: \((6)(-2)(-9) = -12 * -9 = 108\) As we can see, all the equivalent expressions give the same result, 108.

Step 4: Conclusion

By using the commutative property of multiplication, we have found six equivalent expressions for the given problem \((6)(-9)(-2)\), and all the expressions result in the same value, 108. This demonstrates the commutative property of multiplication.

Key Concepts

Equivalent Expressions

When studying algebra, the term equivalent expressions frequently comes up. But what does it mean? Essentially, these are expressions that may look different but represent the same value. For instance, in the given exercise, the expression \( (6)(-9)(-2) \) can be rearranged in many ways due to the commutative property of multiplication. No matter how we change the order of the numbers involved in the multiplication, the result remains a consistent \( 108 \).

This property provides a fundamental understanding that expressions are not just about numbers; they are about relationships that hold true under certain conditions. Recognizing equivalent expressions is crucial in solving algebraic equations, simplifying expressions, and understanding algebraic functions.

Elementary Algebra

Diving into the realm of elementary algebra, we enter a universe where symbols and letters are used to represent numbers and quantities in formulae and equations. It's a step up from basic arithmetic because instead of just numbers, we now deal with variables. A core aspect of elementary algebra is understanding properties of operations, such as the commutative property seen in our textbook exercise.

Understanding how to work with these variables and expressions lays the groundwork for more advanced topics in mathematics and is immensely useful in a variety of real-world contexts, from computing interest to engineering complex systems. It's all about finding unknowns and discovering the connections between different parts of an equation.

Multiplication Properties

The various multiplication properties are the tools that make algebra a lot more manageable. Apart from the commutative property which we've already discussed, there's the associative property that allows us to group numbers in different ways when multiplying, and the distributive property that lets us multiply a number by a sum of numbers efficiently. All these properties simplify complex algebraic manipulations and ensure accuracy. They form an essential part of understanding how multiplication works not just in numbers, but in algebraic expressions as well.

Algebraic Manipulation

When it comes to algebraic manipulation, we are essentially performing operations to rearrange, simplify, or solve algebraic expressions and equations. It's like a craft where one uses the properties of operations, like commutation, association, and distribution, as their tools. The exercise illustrates this concept well by rearranging the multiplication order of numbers using the commutative property, which in turn maintains the equivalence of the expressions.

This skill is particularly important when solving for variables, simplifying expressions, and even in advanced applications like solving simultaneous equations or working with polynomials. Developing a strong foundation in algebraic manipulation is not just about doing well in math class; it prepares students for critical thinking and problem-solving in various scientific and engineering fields.