Question

Determine whether the monomials are like terms. \(T\) and \(-T\)

Short answer

Answer: Yes, the monomials T and -T are like terms because they have the same variables (T) with the same exponents.

Step-by-step solution

Identify the variables in each monomial

Both monomials have the same variable, which is T. There are no additional variables or different exponents.

Compare the variables and exponents

Since both monomials have the same variable, T, and no different exponents or other variables, they are like terms.

Conclusion

The monomials \(T\) and \(-T\) are like terms because they have the same variables (T) with the same exponents.

Key concepts

Monomials

A monomial is an algebraic expression consisting of a single term. It includes coefficients (which are numbers) and variables (like T in our exercise), often attached with exponents. The key to identifying monomials is that they should not have plus or minus signs separating terms. This means expressions like 3x, -7y^2, or 5 are all monomials. In the context of our exercise, both T and -T are monomials. They consist of a single term with no additional plus or minus signs within them.

When working with monomials, the concept of like terms becomes very useful. Like terms are terms that have identical variable parts—meaning the variables and their corresponding exponents must match. Coefficients don't need to match for terms to be considered 'like'—this is crucial to combining or simplifying algebraic expressions. In our exercise, T and -T are indeed like terms, they both have the same variable which is T, with the same exponent, which in this case is 1 (though it is not written explicitly).

Variables in Algebra

Variables are symbols used to represent numbers in algebra. They are often denoted by letters, such as x, y, or T as in our original exercise. Variables are crucial because they allow us to work with numbers that may not be known at the moment, or that can change. We can think of them as placeholders for real numbers.

In an algebraic expression, understanding variables is key to carrying out operations like addition, subtraction, multiplication, and division, as well as more complex functions. When we compare variables in algebra to determine if we have like terms, we look at two main things: whether the variable letter matches and whether any exponents applied to them are the same. For simplicity, if there is no exponent written, it means the exponent is 1. Our task was to check if T and -T are like terms; they are, because the variable T is the same in both.

Exponents

Exponents are a shorthand way to show how many times a number, known as the base, is multiplied by itself. An exponent is written as a small superscript number to the right of the base. For example, in the term 3^4, 3 is the base, and 4 is the exponent, indicating that 3 is multiplied by itself 4 times (3 x 3 x 3 x 3).

In like terms, exponents are key because they must match for the terms to be considered like. If the variables are the same but have different exponents, the terms are not like. For instance, x^2 and x^3 would not be like terms because, despite both having the variable x, the exponents 2 and 3 indicate they involve different orders of multiplication of the variable x. Since T and -T do not have visible exponents, it is implied that the exponent for both is 1. Therefore, these two terms have matching variables and exponents, making them like terms.