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Chapter 3: Problem 272

In the following exercises, translate to an algebraic expression and simplify if possible. The product of \(-4\) and 16

Short Answer

Expert verified
-64

Step by step solution

01

- Understand the Problem

The goal is to translate the given phrase 'The product of \(-4\) and 16' into an algebraic expression.
02

- Translate to Algebraic Expression

The phrase 'the product of' indicates multiplication. Thus, the algebraic expression for 'The product of \(-4\) and 16' is \(-4 \times 16\).
03

- Perform the Multiplication

Now, multiply \(-4\) and 16: \(-4 \times 16 = -64\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplication
Multiplication is a fundamental arithmetic operation that combines groups of equal sizes. When translating words into algebraic expressions, the term 'product' signifies multiplication. For example, in the given exercise, 'The product of \(-4\) and 16' tells us to multiply \(-4\) by 16. The corresponding algebraic expression is \(-4 \times 16\), which we simplify in later steps. Understanding the context and language around multiplication helps translate real-world problems into mathematical expressions.

When working with multiplication:
  • Identify the numbers or variables to be multiplied.
  • Express the multiplication operation using the multiplication symbol '脳' or sometimes a dot '鈥�'.
  • Simplify the resulting expression if possible.
Negative Numbers
Negative numbers are values less than zero, represented with a minus sign (-). They follow certain rules when used in arithmetic operations, such as multiplication and addition. In the given exercise, we use negative number rules in multiplication. When multiplying a negative number with a positive number:
  • The product is negative.
  • Remember the rule: 'A negative times a positive equals a negative.'
Therefore, multiplying \(-4\) by 16 results in \(-64\) because, according to the rule, a negative number multiplied by a positive number remains negative.

Understanding how to handle negative numbers ensures accurate results in mathematical computations and real-world applications.
Simplification
Simplification in mathematics refers to making a problem or expression easier to understand and solve. The goal is to reduce it to its most basic form. After translating the phrase to an algebraic expression, such as \(-4 \times 16\), the next step is to simplify. Here, we perform the multiplication:

\[-4 \times 16 = -64\]

This result \(-64\) is the simplest form of the given problem. By breaking down more complex expressions into simpler components, we make problem-solving more straightforward. To simplify properly:
  • Follow arithmetic rules and properties.
  • Perform operations step-by-step.
  • Check your work to ensure accuracy.
Practicing simplification builds confidence and proficiency in handling various mathematical problems.

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Most popular questions from this chapter

In the following exercises, compare the numbers, using \(<\) or \(>\) or=. (a) \(-5\\_|-5|\) (b) \(-|-2|\\_-2\)

In the following exercises, evaluate each expression. $$ -15-3 u^{2} \text { when } u=-5 $$

In the following exercises, simplify. $$ -15+(-12) $$

In the following exercises, solve each equation using the division property of equality and check the solution. $$ -16 r=-64 $$

In the following exercises, solve. $$ -9 r=-54 $$
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