Problem 23 - Algebra 1: Concepts and Skills

Question

State the degree of the monomial. $$ -100 w^{4} $$

Step-by-Step Solution

Answer

The degree of the monomial \(-100 w^{4}\) is 4.

Step 1: Identify the variables in the monomial

The monomial in question is \(-100 w^{4}\). The variable here is \(w\).

Step 2: Identify the exponent of the variable

In \(-100 w^{4}\), the variable \(w\) is raised to the power of 4.

Step 3: State the degree of the monomial

The monomial's degree is the exponent of the variable. Therefore, the degree of \(-100 w^{4}\) is 4.

Key Concepts

Understanding Monomials

A monomial is a mathematical expression that consists of a single term. This term can be a number, a variable, or a product of numbers and variables. Here are a few key points:

• Monomials do not have any addition or subtraction operations separating terms.
• Examples of monomials include: 5, \(3x^{2}\), and \(-7xy\).
• Monomials are used in algebra to build more complex expressions like polynomials.

In the exercise, \(-100 w^{4}\) is a monomial because it's a single term with a constant and a variable part multiplied together.

Variables in Algebra

In algebra, a variable is a symbol used to represent an unknown value. Usually denoted by letters, such as \(x\), \(y\), or \(w\), variables play a crucial role in forming mathematical expressions and equations.

• Variables are placeholders for values that can change or that may not yet be known.
• In equations, they allow us to solve for unknowns by manipulating the expression.
• In \(-100 w^{4}\), the variable is \('w'\), indicating that \('w'\) can be any number.

Understanding variables is key to grasping how algebra works in problem-solving.

The Role of Exponents

An exponent indicates how many times a base is used as a factor in a multiplication. It's written as a small number to the upper right of the base and plays a vital role in expressing powers in mathematics.

• An exponent of 2, also termed squared, means the base is multiplied by itself (e.g., \(x^2 = x \times x\)).
• Exponents provide a shorthand way to write long repeated multiplication (e.g., \(w^4\) means \(w \times w \times w \times w\)).
• In the monomial \(-100 w^{4}\), the exponent is 4, showing that \('w'\) is multiplied together four times.

The exponent is crucial in determining the degree of a monomial, which is the highest power of the variable present.