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Chapter 1: Problem 8

Identify the base in each expression. $$(2 x)^{4}$$

Short Answer

Expert verified
The base in the expression \( (2x)^4 \) is '2x'.

Step by step solution

01

Understand the structure of the expression

In an expression of the form \( a^n \), 'a' is the base and 'n' is the exponent. The base is the number that is being multiplied, and the exponent indicates how many times the base is used in multiplication.
02

Identify the base

In the expression \( (2x)^4 \), the term inside the brackets, '2x', is the base, because it is what's being multiplied together four times.

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Key Concepts

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

Understanding Exponents
Exponents are a fundamental concept in algebra that reflect how many times a number, known as the "base," is multiplied by itself. For instance, in the expression \(a^n\), the exponent is "\(n\)", dictating the number of times \(a\) (the base) is used in repeated multiplication.

Think of exponents as a shortcut to simplify long multiplication:
  • \(3^2\) implies \(3 \times 3\), equating to 9.
  • Meanwhile, \(5^3\) translates to \(5 \times 5 \times 5\), which equals 125.
Exponents make it easier to handle larger calculations without overcrowded notation, simplifying the way we represent powerful concepts in mathematics.

There are different categories of exponents:
  • Positive exponents: These signify standard multiplication.
  • Negative exponents: Indicate division or reciprocals. For example, \(2^{-3}\) means \(\frac{1}{2^3}\) or \(\frac{1}{8}\).
  • Zero exponents: Any non-zero number raised to the power of 0 results in 1, such as \(7^0 = 1\).
Base Identification
Identifying the base in algebraic expressions is crucial for understanding how the expression is structured. In an expression like \((2x)^4\), the "base" is the component inside the parentheses. Here, "2x" acts as the base.

The role of the base can be understood with these key points:
  • The base is the number or expression that is multiplied by itself.
  • It directly influences the outcome when combined with an exponent.
Examples clarify the idea:
  • In \((3)^2\), the base is 3, leading to \(3 \times 3\).
  • In the more complex \((2x)^4\), the base is "2x," resulting in \( (2x) \times (2x) \times (2x) \times (2x) \).
Recognizing the base ensures proper manipulation and simplification, a necessary step in solving or transforming algebraic expressions effectively.
Deciphering Mathematical Expressions
Mathematical expressions represent operations, quantities, and variables. They form the backbone of solving algebraic problems and expressing equations.

Understanding them means knowing how to interpret and manipulate the components present in these expressions.
  • Terms: Components separated by a '+' or '-' sign. Each term includes numbers and/or variables, sometimes governed by exponents.
  • Factors: Elements being multiplied together within a term.
  • Operations: Basic functions such as addition, subtraction, multiplication, and division that dictate how to approach an expression.
In the expression \((2x)^4\), it's essential to identify:
  • "2x" as a term and base.
  • "4" as the exponent dictating the repetition of multiplication.
  • The operation of exponentiation indicating multiplication of the base repeatedly.
These basic insights help in simplifying, solving, or rearranging mathematical expressions effectively. By mastering how to work with these expressions, students become more adept at tackling higher-level math problems.

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Most popular questions from this chapter

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