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Chapter 5: Problem 58

Simplify. $$ (3 a b c) 3 $$

Short Answer

Expert verified
The simplified expression is \(9abc\).

Step by step solution

01

Identify the Terms

The expression given is \((3abc)3\). This consists of the term \(3abc\) being multiplied by another constant, \(3\).
02

Multiply the Constants

Multiply the numeric coefficients. Here, we have the coefficient from \(3abc\), which is \(3\), and the standalone \(3\). Multiply them: \(3 \times 3 = 9\).
03

Combine the Variables

Since \(abc\) is being multiplied by the constant \(9\), the variable part stays the same. The expression is now \(9abc\).

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Key Concepts

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

Multiplication of Constants
When simplifying an algebraic expression, one of the initial steps is the multiplication of constants. In our example, the expression given is \((3abc)3\). Here, we have two constants, both of which are 3, located in different parts of the expression. To simplify, we need to multiply these constants together.

  • The constants in this scenario are the numbers that precede the variables, which are numeric coefficients.
  • These coefficients are multiplied just like regular numbers.
  • In our case, multiplying the constants means calculating: \(3 \times 3 = 9\).
Understanding this multiplication is crucial as it helps simplify the expression further. The numerical value derived from this multiplication will later work hand in hand with the variable component.
Combining Like Terms
"Combining like terms" involves joining together terms in an expression that have identical variable parts. In the context of our example, it's important to note that although we do not have multiple terms with variables to directly combine, recognizing this concept helps explain why the expression simplifies smoothly after multiplying constants.

  • Terms are considered "like" when they share the same variables with identical exponents.
  • In this exercise, the variable "bucket," \(abc\), remains consistent throughout the expression.
  • The result after multiplication becomes \(9abc\), maintaining the original arrangement of variables.
Although we haven't combined different terms here, acknowledging this concept provides a foundational understanding crucial for tackling more complex expressions.
Algebraic Expressions
Algebraic expressions are combinations of variables, constants, and arithmetic operations (like addition, subtraction, and multiplication). In our example, the expression \((3abc)3\) involves both numbers and letters representing variables. Simplifying this expression involves reducing it to its simplest form while retaining the meaning.

For an efficient approach:
  • Identify the constants and variables within the expression.
  • Apply arithmetic operations to the constants, while noting the composition of the variables.
  • The objective is to achieve a form where terms are neatly organized, such as \(9abc\).
Grasping the structure of algebraic expressions is valuable. This knowledge allows you to manipulate and simplify expressions seamlessly. It's also a stepping stone to understanding more complex algebraic concepts.

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