Problem 66 - Intermediate Algebra

Question

Explain how the associative and commutative properties can help simplify \([(25)(97)](-4)\).

Step-by-Step Solution

Answer

The simplified result is -9700.

Step 1: Understanding the Problem

We need to simplify the expression \([(25)(97)](-4)\). The expression involves multiplication of three numbers: 25, 97, and -4.

Step 2: Recall the Properties

The associative property of multiplication states that the grouping of numbers does not change the product, i.e., \((a imes b) imes c = a imes (b imes c)\). The commutative property states that the order of numbers can be changed in multiplication, i.e., \(a imes b = b imes a\).

Step 3: Apply the Commutative Property

We can use the commutative property to reorder the multiplication: \( (25)(97)(-4) = (-4)(25)(97) \). This reordering can help make calculations simpler based on the numbers involved.

Step 4: Apply the Associative Property

Using the associative property, we can regroup: \((-4)(25)(97) = (-4 imes 25)(97)\). Calculate \(-4 imes 25 = -100\).

Step 5: Calculate the Product

Now, we have \((-100)(97)\). Calculate this product: \(-100 imes 97 = -9700\).

Step 6: Verify the Solution

Verify by calculating differently, e.g., \((25)(97) = 2425\) and then \((2425)(-4) = -9700\). Both methods give the same result, confirming correctness.

Key Concepts

Multiplication

Multiplication is a fundamental mathematical operation that involves combining numbers to get a total amount. In the expression \([(25)(97)](-4)\), multiplication is used to find the product of the numbers 25, 97, and -4. Each number in the set is called a factor, and together they yield a product.
In essence, multiplication can be seen as repeated addition, but it is a faster and more efficient way to handle multiple additions of the same number. For example, multiplying 25 by 97 can be compared to adding 25 a total of 97 times.
When dealing with negative numbers such as -4, multiplication also involves the rules for positive and negative numbers, where a positive number times a negative number results in a negative product.

Algebra

Algebra allows us to solve problems using symbols and letters to represent numbers and quantities in mathematical expressions. It provides a way to structure and solve problems that involve unknown values.
In our specific exercise with \([(25)(97)](-4)\), algebra is used to apply mathematical properties like commutative and associative properties to simplify expressions. By understanding these properties, we can rearrange and group factors to make calculations more straightforward.
This use of algebraic thinking is vital for expressing and solving more complex mathematical problems, and it gives you tools to systematically approach equations and expressions.

Simplification

Simplification in mathematics refers to the process of reducing expressions or equations to their simplest form. In the given problem, we aim to make computations easier by applying mathematical properties.
For example, by applying the commutative property, we reorder the factors in \([(25)(97)](-4)\) to \((-4)(25)(97)\). This sets the stage for simplification through the associative property, where we group and calculate \(-4 \times 25\) first to get \-100\, then multiply by 97.
The goal of simplification is not just to find an answer, but to transform the expression into a form that is easier to handle and understand.

Mathematical Properties

Mathematical properties such as associative and commutative properties serve as essential tools in simplifying and solving expressions. These properties define how numbers behave under certain operations like addition and multiplication.
**Commutative Property** allows us to change the order of multiplication or addition without changing the result: \(a \times b = b \times a\). This was used to reorder \(25 \times 97 \times -4\) into a more convenient order for multiplication.
**Associative Property** allows us to change the grouping of numbers: \((a \times b) \times c = a \times (b \times c)\). This was used to group \((-4 \times 25) \times 97\) so we could simplify calculation steps.
These properties are powerful because they make our calculations more flexible and adaptable, helping us find the solution more efficiently.