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Chapter 7: Problem 75

In the following exercises, simplify. $$ -23(2 n) $$

Short Answer

Expert verified
-46n

Step by step solution

01

- Identify the Expression

Recognize that the expression needs to be simplified: \(-23(2n)\)
02

- Apply the Distributive Property

Multiply the factor outside the parentheses by the term inside the parentheses: \(-23 \times 2n\)
03

- Multiply the Numbers

Calculate the product of \(-23\) and \(2\): \(-23 \times 2 = -46\)
04

- Combine the Variable

Attach the variable \(n\) to the product from the previous step: So, \(-46n\) is the simplified form.

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Key Concepts

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

Distributive Property
The distributive property is an essential tool in algebra. It allows you to multiply a single term across terms within parentheses. This property states that for any numbers or variables, \(a(b + c) = ab + ac\). It means you distribute the multiplication to each term inside the parentheses. In our exercise, we use the distributive property to simplify \(-23(2n)\). We multiply \(-23\) by each part inside the parentheses. This makes the expression easier to work with. Understanding this property is crucial for simplifying complex algebraic expressions. Examples include:
  • \(3(a + b) = 3a + 3b\)
  • \(-5(x - y) = -5x + 5y\)
Applying this property is often the first step in simplifying an algebraic expression.
Multiplication of Integers
When multiplying integers, you must consider both the number values and the signs. Positive times positive gives a positive result. Positive times negative, or negative times positive, yields a negative result. Negative times negative results in a positive outcome. So in our problem, we multiply \(-23\) (a negative integer) by \(2\) (a positive integer). The multiplication steps are straightforward:
  • Ignore the signs and multiply the number values: \(23 \times 2 = 46\)
  • Determine the sign of the product. Since one integer is negative, the result is negative.
Therefore, \(-23 \times 2 = -46\). This multiplication result helps us continue simplifying the original expression.
Combining Like Terms
Combining like terms is about simplifying expressions by merging terms that have identical variables. Only terms with the same variable part can be combined. In our exercise, once we have simplified the multiplication, we are left with \(-46n\). This term cannot be further simplified since there are no other like terms given. To combine like terms, follow these steps:
  • Identify terms with the same variable.
  • Add or subtract their coefficients.
Let's look at a more detailed example:
  • Consider \(-3n + 2n - n\).
  • All terms have the same variable part, \(n\).
  • Add their coefficients: \(-3 + 2 - 1 = -2\).
  • Thus, the simplified expression is \(-2n\).
Understanding how to combine like terms is fundamental for simplifying more complex algebraic expressions in your studies.

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Most popular questions from this chapter

In the following exercises, evaluate each expression for the given value. If \(k=65,\) evaluate: $$ (a)\frac{4}{9}\left(\frac{9}{4} k\right) $$ $$ (b)\left(\frac{4}{9} \cdot \frac{9}{4}\right) k $$

In the following exercises, convert between U.S. units. Round to the nearest tenth. The play lasted 1\(\frac{3}{4}\) hours. Convert the time to minutes.

In the following exercises, simplify. $$ 0 \div \frac{2}{3} $$

In the following exercises, find the multiplicative inverse. $$ \frac{1}{10} $$

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