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Chapter 4: Problem 32

Identify the base and the exponent in each expression. $$(-3 x)^{2}$$

Short Answer

Expert verified
In the expression \((-3x)^2\), the base is \(-3x\) and the exponent is 2.

Step by step solution

01

Identify the Base

Look for the number or expression that is being raised to a power; this is the base. In this expression, the base is the quantity inside the parentheses, \(-3x\). So, the base of \((-3x)^2\) is \(-3x\). Remember that the base includes both the negative sign and the variable \(x\).
02

Identify the Exponent

The exponent is the number that tells how many times the base is used as a factor. In this expression, the exponent is the number outside the parentheses, which is 2. So, the exponent of \((-3x)^2\) is 2.

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Key Concepts

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

Base of an Expression
In any expression of the form \(a^n\), the term \(a\) is referred to as the base. The base is what you are multiplying by itself a certain number of times, as dictated by the exponent \(n\). Simply put, the base is the number or expression being
  • raised to a power, which means it's the number you multiply repeatedly.
  • It can be a single number, a variable, or a combination of both.
  • For example, in the expression \((-3x)^2\), the base is the entire quantity inside the parentheses, \(-3x\).

Remember, when the base is a combination that involves more than just a simple number, such as including parentheses or a negative sign with variables, everything within the grouping symbol is the base.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and at times operators like addition or multiplication.
  • Variables are symbols, usually letters, that stand for unknown values.
  • Algebraic expressions can range from very simple, like \(x + 2\), to complex expressions like \((-3x)^2 + 5x - 7\).
  • Such expressions help us solve for variables and understand real-world phenomena through mathematical equations.


Algebraic expressions can also be plugged into equations, allowing you to solve for unknowns. It's essential to learn how to manipulate these expressions accurately by understanding how elements such as exponents and bases work together in expressions. Moreover, becoming familiar with identifying and simplifying algebraic expressions is a foundational part of working in algebra.
Negative Base
A negative base is a base that includes a negative sign. When you have an expression like \((-3x)^2\), the negative sign is as crucial as any other part within the base.
  • In expressions with a negative base, pay close attention to parentheses. They indicate that the base includes not just the variable or number, but also the negative sign.
  • Applying an exponent to a negative base can change results depending on whether the exponent is even or odd.
  • For example, \((-3x)^2\) becomes positive because the square of any negative number makes it positive.
Without properly understanding the negative base, it becomes easy to make mistakes in calculations and assumptions, leading to incorrect results. Keeping track of whether your base is positive or negative will greatly affect how your final answers come out, especially as you work with more complex expressions.

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Most popular questions from this chapter

Describe how to determine the degree of a polynomial.

Let \(P(x)=3 x-5 .\) Find each value. $$P(x+h)-P(x)$$

Graph each polynomial function. State the domain and range. $$f(x)=x^{3}+2$$

Classify each polynomial as a monomial, a binomial, a trinomial, or none of these. $$x^{3}$$

Simplify or solve as appropriate. $$4+(2 y-3)^{2}=(2 y-1)(2 y+3)$$
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