Problem 26 - Algebra for College Students

Question

Simplify each numerical expression. Be sure to take advantage of the properties whenever they can be used to make the computations easier. $$(2)(17)(-5)-(4)(13)(-25)$$

Step-by-Step Solution

Answer

The simplified expression is 1130.

Step 1: Simplify Each Group

First, simplify the multiplication within each parenthesis: 1. For the first group \( (2)(17)(-5) \): - Multiply \(2\) and \(17\): \[ 2 \times 17 = 34 \] - Then multiply the result by \(-5\): \[ 34 \times (-5) = -170 \] 2. For the second group \( (4)(13)(-25) \): - Multiply \(4\) and \(13\): \[ 4 \times 13 = 52 \] - Then multiply the result by \(-25\): \[ 52 \times (-25) = -1300 \]

Step 2: Substitute and Simplify

Now substitute the simplified results back into the expression:\( -170 - (-1300) \)This simplifies to \( -170 + 1300 \).

Step 3: Perform Addition

Add \(-170\) and \(1300\):\[ -170 + 1300 = 1130 \]

Step 4: Verification

Verify the correctness of the solution by checking each computational step for accuracy: - The multiplication within the groups was confirmed. - Substitution corrected for double negatives. - Addition verified. The solution matches logical expectations.

Key Concepts

Properties of Operations

Understanding the properties of operations is crucial in simplifying numerical expressions efficiently. These properties include the commutative, associative, and distributive properties.

• Commutative Property: This property states that the order of addition or multiplication does not matter. For example, in multiplication, \( a \times b = b \times a \).
• Associative Property: This means that the way you group numbers does not change the result. In multiplication, \( (a \times b) \times c = a \times (b \times c) \).
• Distributive Property: This property connects multiplication and addition, showing that \( a \times (b + c) = a \times b + a \times c \). However, our current problem being purely multiplication does not apply this directly.

Applying these properties allows you to rearrange and group numbers in any way that simplifies calculations. For the given numerical expression, even though the properties don’t seem directly used, the order of multiplication and management of negative signs are influenced by these foundational properties. Knowing that multiplication is both commutative and associative helps us understand why these operations can be executed seamlessly in different orders.

Multiplication of Integers

Multiplying integers is straightforward, especially when you remember the rules for handling positive and negative numbers. Let’s review these rules:

• Positive Times Positive: The result is positive. For example, \( 3 \times 4 = 12 \).
• Negative Times Negative: Multiplying two negative numbers results in a positive number, like \( -3 \times -4 = 12 \).
• Positive Times Negative or Negative Times Positive: The result will always be negative. For instance, \( 3 \times -4 = -12 \).

In our original exercise, understanding these rules allowed us to break down the expression efficiently. Multiplying \(2 \times 17\) we got \(34\), and then with \(-5\), the result was \(-170\). Similarly, multiplying \(4 \times 13\) yielded \(52\), and then with \(-25\), it became \(-1300\). These steps show how recognizing the sign rules helps anticipate whether the result will be positive or negative.

Addition of Integers

Adding integers requires a methodical approach, especially when dealing with both positive and negative numbers. The following are the key rules:

• Positive Plus Positive: This is simple addition, like \(3 + 2 = 5\).
• Negative Plus Negative: Combine their absolute values, and the result is negative. For instance, \(-3 + -4 = -7\).
• Positive Plus Negative or Negative Plus Positive: Subtract the smaller absolute value from the larger absolute value and the result takes the sign of the number with the larger absolute value. For example, \(5 + (-3) = 2\) or \(-5 + 3 = -2\).

In the given exercise, handling the transition from subtraction to addition is crucial. The task was \(-170 - (-1300)\). By recognizing that subtracting a negative is the same as adding a positive, the original problem reduces to \(-170 + 1300\). This results in the final answer, \(1130\), showing not just computation but understanding of integer addition rules.