Problem 20 - 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. $$ (14)(25)(-13)(4) $$

Step-by-Step Solution

Answer

The expression simplifies to -18200.

Step 1: Group and Rearrange Terms

We start by grouping and rearranging the numbers to simplify our multiplication using the commutative property of multiplication, which states that numbers can be multiplied in any order. Let's rearrange the terms as follows: \[ (14)(4)(25)(-13) \]

Step 2: Simplify Grouped Terms

Next, simplify the grouped terms. Start by multiplying the two positive numbers together to make computations easier: \[ 14 \times 4 = 56 \] Therefore, the expression becomes: \[ (56)(25)(-13) \]

Step 3: Further Simplify by Multiplying Next Pair

Now, multiply 56 by 25: \[ 56 \times 25 = 1400 \]Now, the expression becomes: \[ 1400 \times (-13) \]

Step 4: Multiply with the Negative Number

Finally, multiply 1400 by -13:\[ 1400 \times (-13) = -18200 \] Note that the result is negative because we are multiplying by a negative number.

Key Concepts

Properties of Multiplication

Multiplication is an essential operation in mathematics that comes with several helpful properties. These properties can make complicated calculations a lot easier and faster to solve.

When multiplying numbers, there are a few key properties to keep in mind:

• Associative Property: This property states that the way in which factors are grouped in multiplication does not affect the product. In simpler terms, it means that you can group numbers in any order, and the answer will be the same. Algebraically, this is expressed as \( (a \times b) \times c = a \times (b \times c) \).
• Distributive Property: This property connects multiplication and addition or subtraction. If you need to simplify an expression like \(a(b+c)\), you can multiply both terms inside the parentheses by \(a\). Mathematically, this is written as \( a \times (b + c) = a \times b + a \times c \).

Using these properties strategically can save you time and make calculations less prone to mistakes.

Commutative Property

The commutative property is an easy-to-understand rule that states numbers can be multiplied in any order without changing the result. This means if you're given a multiplication problem, you can rearrange the numbers in any way that makes it simpler to solve.

For instance, in the expression \((14)(25)(-13)(4)\), using the commutative property allows us to rearrange to \((14)(4)(25)(-13)\). By doing this, you can first multiply numbers that are easier to calculate together, like \(14 \times 4\).

• The mathematical expression for the commutative property is \(a \times b = b \times a\).
• Using this property helps simplify the operations and identify easier calculations we can perform first.

Taking advantage of this property, especially when dealing with multiple numbers, can significantly simplify the multiplication process.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, often making use of different mathematical properties. The goal is to make calculations more straightforward and find the solution effectively. It often involves performing operations in a strategic order to simplify the work needed.

In the exercise, we started with a complex expression \((14)(25)(-13)(4)\). By rearranging and breaking it down step by step:

• We initially rearranged to \((14)(4)(25)(-13)\), enabling an easier initial multiplication: \(14 \times 4 = 56\).
• This rearrangement led us to \((56)(25)(-13)\) which was further simplified with multiply \(56 \times 25 = 1400\).

The systematic simplification helps not only to reach the final answer quickly but also ensures you avoid errors in the calculation process.

Multiplication with Negative Numbers

Multiplying with negative numbers often trips people up, but it's straightforward once you understand the rule: multiplying a positive number by a negative number results in a negative product. This rule is important to keep in mind whenever you have a mix of positive and negative numbers.

Here's how it applies to our case:

• When we got to the expression \(1400 \times (-13)\), the rule tells us the product will be negative.
• Multiplying these gives us \(-18200\).

Remember:- Two positive numbers make a positive product.- Two negative numbers also make a positive product (because a negative times a negative is positive).- A positive number with a negative number makes a negative product.
Understanding these rules helps manage signs during multiplication, ensuring you get the correct product sign once calculations are complete.