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Chapter 1: Problem 29

Simplify each expression. $$ 5(4 c)(3) $$

Short Answer

Expert verified
The simplified expression is 60c.

Step by step solution

01

Rearrange Multiplication

First, notice that the expression consists of three terms being multiplied together: 5, 4c, and 3. Remember, multiplication is commutative, so you can rearrange the terms in any order to make it easier to simplify. We will rearrange it as (5)(4)(c)(3).
02

Multiply the Numerical Terms

Multiply the numbers: (5)(4)(3). Begin by multiplying 5 and 4, which equals 20. Then multiply this result by 3: (20)(3) = 60.
03

Attach the Variable Term

Now, take the result of the multiplication of the numbers (60) and multiply by the variable term c: (60)(c) = 60c.

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Key Concepts

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

Commutative Property
The commutative property is a fundamental concept in algebra that states the order of numbers in a multiplication statement doesn't affect the product. In simpler terms, this means that no matter how you arrange the numbers being multiplied, the result will be the same. For instance, in the expression \(5 \times 4 \times 3\), you could calculate it as \((5 \times 4) \times 3\) or rearrange it like \((4 \times 3) \times 5\). Either way, you'll end up with the same result.
  • Comes in handy to simplify complex expressions.
  • Allows flexibility in calculating by changing the order of factors.
Using the commutative property, one can find more efficient ways to multiply large or awkward numbers, simply by rearranging their order.
Multiplication of Terms
When simplifying algebraic expressions, understanding how to multiply terms is key. Multiplying terms involves combining numerical coefficients first and then appending any variables afterward. Consider the expression \( 5 \times 4 \times (c) \times 3 \). Start with the numerical multiplication:
  • First multiply the numbers: \(5 \times 4 = 20\).
  • Next, multiply your result by the remaining number: \(20 \times 3 = 60\).
Once you've calculated the product of the numbers, leave the variable part for last. Attach the variable term to complete the simplification without changing its meaning. In this example, taking 60 and adding \(c\) results in \(60c\).Multiplying terms become straightforward: numerical coefficients first, variables later.
Variable Manipulation
Variable manipulation is about handling and simplifying expressions that include variables. Variables are symbols, often letters, representing numbers in mathematical expressions. In algebra, you'll frequently encounter problems that require variable manipulation. This concept involves a few critical skills:
  • Recognizing how variables interact with numerical values.
  • Understanding that variables, when multiplied by numbers, act as placeholders to build expressions.
For the expression \(60 \times c\), the number 60 is multiplied by the variable \(c\). This operation implies any number that \(c\) represents will be multiplied by 60, making the entire expression \(60c\). Here, variable manipulation simplifies an expression by clearly showing the relationship between different mathematical components.

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Most popular questions from this chapter

The commutative property states that changing the order when multiplying does not change the answer. Are the following activities commutative? Explain. a. Washing a load of clothes; drying a load of clothes b. Putting on your left sock; putting on your right sock

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