Problem 15 - Elementary and Intermediate Algebra

Question

Identify the base and the exponent in each expression. A. \((-3 x)^{2}\) B. \(-3 x^{2}\) C. \(-(-3 x)^{2}\)

Step-by-Step Solution

Answer

A: Base = \(-3x\), Exponent = 2; B: Base = \(x\), Exponent = 2; C: Base = \(-3x\), Exponent = 2.

Step 1: Understanding the Expression A

The expression given is \((-3x)^2\). The number or variable inside the parentheses that is raised to a power is called the base. Here, the base is \(-3x\). The number to which the base is raised is the exponent, which is 2 in this case.

Step 2: Breaking Down Expression B

In the expression \(-3x^2\), the exponent applies only to the \(x\), so the base is the \(x\) and the exponent is 2. The \(-3\) is a coefficient and is not part of the base.

Step 3: Analyzing Expression C

The expression is \(-(-3x)^2\). The parentheses indicate that \(-3x\) is considered together as the base. The minus sign outside the parentheses does not affect the determination of the base or exponent inside. Thus, the base is \(-3x\) and the exponent is 2.

Key Concepts

Base Identification

When dealing with exponentiation, identifying the base is crucial for understanding and solving expressions. The base is the number or algebraic term that is being multiplied by itself. In expressions like \((-3x)^2\), \(-3x\) inside the parentheses is the base. To identify the base:

• Look for the term that the exponent is associated with. This is often enclosed in parentheses or directly preceding the exponent.
• If parentheses are present, everything inside them is the base.
• Coefficients (like \(-3\) in \(-3x^2\)) outside a variable without parentheses are not part of the base.

Understanding this can greatly simplify solving algebraic expressions, ensuring the right component is utilized in calculations.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations all combined. These expressions can have constants (like \(-3\)), variables (like \(x\)), and a variety of operations such as addition or multiplication.Key components to remember:

• Variables: These are symbols, commonly letters, that represent unknown values. They can change and interact with numbers in operations.
• Coefficients: These are the numbers in front of variables (e.g., \(-3\) in \(-3x\)) which multiply the variable.
• Operations: Such as addition, subtraction, multiplication, and exponentiation, dictate how the terms in the expression interact.

Recognizing each part of the expression helps clarify its structure and how to solve or manipulate it effectively.

Algebra Techniques

Various algebraic techniques are used to manipulate and solve equations or expressions. These techniques include simplifying expressions, expanding terms, and factoring.Some common techniques include:

• Simplifying: Combining like terms (terms with the same variable and exponent) e.g., simplifying \(2x + 3x = 5x\).
• Expanding: Distributing a term across terms in parentheses, such as \(a(b + c) = ab + ac\).
• Factoring: Rewriting an expression as a product of its factors, like turning \(x^2 + 5x + 6\) into \((x + 2)(x + 3)\).

Utilizing these techniques helps in solving complex problems and understanding the relationships between terms more clearly. Each tool aids in breaking down expressions into more manageable parts.