Problem 19 - Intermediate Algebra

Question

Simplify each numerical expression. Be sure to take advantage of the properties whenever they can be used to make the computations easier. $$ (25)(-13)(4) $$

Step-by-Step Solution

Answer

The simplified expression is \(-1300\).

Step 1: Understand the Expression

The expression to simplify is \((25)(-13)(4)\), which involves multiplying three numbers together.

Step 2: Use the Commutative Property

The Commutative Property of Multiplication states that numbers can be multiplied in any order. Reorder the numbers to make multiplication simpler: \((25 \times 4) \times (-13)\).

Step 3: Perform the Multiplication of Two Terms

Multiply the first two numbers: \(25 \times 4 = 100\). This gives us the simplified expression \(100 \times (-13)\).

Step 4: Multiply by the Negative Number

Now multiply the result by \(-13\): \(100 \times (-13) = -1300\). The product of \(100\) and \(-13\) is \(-1300\).

Step 5: Conclusion

After applying the properties of multiplication and performing the calculations, the simplified expression is \(-1300\).

Key Concepts

Commutative Property of Multiplication

The commutative property of multiplication is a fundamental math rule that simplifies our lives. This property tells us that the order in which you multiply numbers does not change the product. For example, if you have two numbers, such as 3 and 5, you can multiply them as \(3 \times 5\) or \(5 \times 3\), and the result will still be 15. This property applies to any number of numbers you multiply.

What's great about this property is that it can make complex calculations much simpler. Imagine you have a string of numbers to multiply, such as in our example with \((25)(-13)(4)\). By using the commutative property, you can rearrange these numbers to multiply the easiest pair first. This way, you can break down a tough problem into simple, manageable steps.

Properties of Multiplication

Multiplication has several properties that help make computations more manageable. Aside from the commutative property, there are a few other key properties to know:

• Associative Property: This property permits us to change the grouping of numbers being multiplied. For instance, \((a \times b) \times c = a \times (b \times c)\). This is helpful when dealing with multiple numbers.
• Identity Property: This is one of the simplest properties. It states that any number multiplied by 1 remains unchanged. For example, \(a \times 1 = a\).
• Zero Property: Multiplying any number by zero results in zero. For example, \(a \times 0 = 0\). This property is important because it can simplify problems quickly.

In the exercise, we utilized the commutative property to rearrange and multiply \(25\) and \(4\) first, making use of these multiplication properties can greatly simplify our calculations.

Multiplying Negative Numbers

Multiplying negative numbers can seem confusing at first, but with a bit of practice, it becomes straightforward. When you multiply a positive number by a negative number, the result is always negative. A useful way to remember this is that one negative makes the whole expression negative.

For instance, in our example, we needed to multiply \(100\) by \(-13\). Since \(100\) is positive and \(-13\) is negative, the product is negative. Thus, \(100 \times (-13) = -1300\).

If you multiply two negative numbers together, the result becomes positive. Think of it as two negatives "canceling out" each other. Always remember:

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

Mastering these rules will help you accurately solve multiplication problems involving negative numbers.