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Chapter 1: Problem 10

Complete each statement. The numerical coefficient in the term \(-7 y z^{2}\) is

Short Answer

Expert verified
The numerical coefficient is -7.

Step by step solution

01

Identify the Term

Examine the term given in the problem: -7 y z^{2}.
02

Understand the Term Components

The term consists of three parts: the numerical coefficient, the variable y, and the variable z raised to the power of 2.
03

Locate the Numerical Coefficient

The numerical coefficient in a term is the number that multiplies the variables. In the term -7 y z^{2}, the numerical coefficient is -7.

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Key Concepts

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

Algebraic Terms
An algebraic term is a single mathematical expression that can include constants, variables, and exponents. Each term is separated by plus or minus signs in an equation. For example, in the expression \( -7 y z^{2} \), \( -7y z^{2} \) is a single algebraic term.

Algebraic terms have specific parts:
  • **Numerical Coefficient:** This is the constant number that multiplies the variables.
  • **Variables:** These are symbols that represent numbers, typically letters like \( y \) and \( z \).
  • **Exponents:** These indicate how many times a variable is multiplied by itself.
Understanding algebraic terms is crucial for simplifying and solving algebraic expressions. Each part of the term serves a unique purpose and helps in manipulating equations correctly.
Variables
Variables are symbols that stand for unknown values and allow algebraic expressions to be generalized. In the term \( -7 y z^{2} \), the variables are \( y \) and \( z \). They can represent any numerical value, making it possible to solve diverse mathematical problems. Variables function as placeholders and enable us to write equations that model real-world scenarios.

In algebra:
  • **Independent Variables:** These are the variables whose values are chosen without considering other variables.
  • **Dependent Variables:** These variables depend on the values of other variables in the equation.
For instance, if we have an equation involving \( y \) and \( z \), we might assign different values to see how they affect the outcome. Recognizing variables in terms helps in performing operations like addition, subtraction, multiplication, and division.
Exponents
Exponents denote how many times a number (the base) is multiplied by itself. In the term \( -7 y z^{2} \), \( z^{2} \) indicates that \( z \) is multiplied by itself once, or \( z \times z \). Understanding exponents is important for working with algebraic expressions and equations.

Key points about exponents:
  • **Base:** The number or variable that is being multiplied.
  • **Exponent:** The small, raised number indicating how many times the base is used as a factor.
  • **Standard Form:** When written in standard form, an exponent expresses repeated multiplication.
An exponent simplifies expressions and can be positive, negative, or even fractional, expanding the range of problems you can solve using algebra.

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Most popular questions from this chapter

Decide whether each statement is true or false. If it is false, correct the statement so that it is true. \(\sqrt{25}\) is a positive number.

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