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Chapter 1: Problem 148

Write an equivalent expression using the distributive law. $$ -2(3 c+5 d) $$

Short Answer

Expert verified
-6c - 10d

Step by step solution

01

Identify the distributive law

The distributive law states that for any numbers, variables, or expressions, distributive property: \[ a(b + c) = ab + ac \]
02

Apply the Distributive Property to the Expression

Use the distributive property to distribute \( -2 \) to each term inside the parentheses. This means multiplying \( -2 \) by both \( 3c \) and \( 5d \) resulting in: \[ -2(3c + 5d) = -2 \times 3c + (-2) \times 5d \]
03

Perform the Multiplication

Multiply \( -2 \) by each term: \(-2 \times 3c = -6c \) \(-2 \times 5d = -10d \).
04

Write the Final Expression

Combine the results of the multiplication to form the final equivalent expression: \[ -6c - 10d \]

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are a fundamental part of mathematics. They consist of numbers, variables, and arithmetic operations. In our exercise, the expression is \( -2(3c + 5d) \). This includes the constant -2, and the terms 3c and 5d.

Understanding algebraic expressions helps in decoding problems and finding solutions. When you see an expression like this, remember it represents a combination of numbers and variables reflecting a mathematical relationship.

This relationship is then simplified or evaluated using various algebraic rules and properties, such as the distributive property which we will discuss below.
Multiplication
Multiplication in algebra is more than just basic arithmetic. It involves combining numbers and variables to find a product. In our exercise, multiplication is used to apply the distributive property.

Here, we multiply -2 by each term inside the parentheses: 3c and 5d. This step is crucial as it spreads or distributes the -2 across both terms.

Specifically, we perform these operations:
\( -2 \times 3c = -6c \)
\( -2 \times 5d = -10d \).

This multiplication simplifies the original expression by breaking it down into smaller, easier-to-manage parts.
Variable Terms
Variable terms are parts of an algebraic expression that include variables, like c or d, combined with numbers. In our example, 3c and 5d are variable terms.

These terms are affected by the multiplication and distributive steps, as we saw with the original expression \( -2(3c + 5d) \).

Understanding how to manipulate these terms is key in algebra. By applying the distributive property and performing multiplication, we've transformed the original terms into \( -6c \) and \( -10d \).

This manipulation helps us create equivalent expressions that are often simpler and more useful for solving algebraic problems.

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Most popular questions from this chapter

Which numbers in the list provided are (a) whole numbers? (b) integers? (c) rational numbers? (d) irrational numbers? (e) real numbers?. $$ -6.08,-5,0,1, \sqrt{17}, \frac{99}{2} $$

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ 2^{2}-5^{2} $$

Calculate using the rules for order of operations. $$ 17-\sqrt{11-(3+4)} \div[-5-(-6)]^{2} $$

Explain in your own words why \(7 / 0\) is undefined.

Solve and check. The symbol \(\sqsubseteq\) indicates an exercise designed to be solved with a calculator. $$ 6 x-\\{5 x-[7 x-(4 x-(3 x+1))]\\}=6 x+5 $$
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