Question

Determine whether the monomials are like terms. \(4 n^{2}\) and \(7 n^{2}\)

Short answer

Answer: Yes, the monomials \(4 n^{2}\) and \(7 n^{2}\) are like terms.

Step-by-step solution

Identify the variables and exponents

In the first monomial, \(4 n^{2}\), the variable is \(n\) and the exponent is \(2\). In the second monomial, \(7 n^{2}\), the variable is also \(n\) and the exponent is \(2\).

Compare the variables and their exponents

Now that we have identified the variables and their exponents for both monomials, let's compare them. For the first monomial, we have \(n\) with an exponent of \(2\). For the second monomial, we also have \(n\) with an exponent of \(2\). Since the variable and exponent in both monomials are the same, these terms are like terms. Therefore, the monomials \(4 n^{2}\) and \(7 n^{2}\) are like terms.

Key concepts

Monomials

In algebra, a monomial is a mathematical expression that consists of a single term. To break it down, it has one coefficient and one or more variables raised to non-negative integer powers. For example, in the expression 4 n^{2} , 4 is the coefficient, and n^{2} is the variable raised to the power of 2. Monomials are the building blocks of polynomials, which are sums of multiple monomials.

When we deal with monomials, we often need to determine if they are like terms, which means they have the same variables raised to the same exponents. This is important because like terms can be combined through addition or subtraction. For instance, 4 n^{2} and 7 n^{2} are like terms because the variable portions n^{2} match exactly. It’s essential to recognize like terms to simplify algebraic expressions effectively.

Variables in Algebra

In algebra, variables are symbols used to represent unknown values. They are usually denoted by letters such as x, y, z, n , etc. Variables can be thought of as placeholders for numbers that we are either searching for or that can vary. In the equation 4 n^{2} , n is the variable.

Understanding variables is crucial for working with algebraic expressions, equations, and functions. When examining multiple expressions, like in our exercise, the consistent use of a variable—like n—allows us to draw direct comparisons between them. If the variables and their associated exponents are the same in two expressions, this indicates the potential for further algebraic manipulation, such as combining like terms.

Exponents

An exponent in algebra refers to the power to which a number or variable is raised. It is a concise way to represent repeated multiplication of the same factor. In the monomial 4 n^{2} , the exponent is 2, which means that n is multiplied by itself once. The exponent is written as a superscript.

Understanding the role of exponents is key in determining like terms. For example, n^{2} in 4 n^{2} and 7 n^{2} indicates that n is squared in both terms. If the bases (the variables) and their exponents are the same, then the monomials are considered like terms. When working with exponents, remember the basic rules, such as any number to the zero power is 1, and the same bases with exponents can be added or subtracted if they have like terms.