Problem 85 - Elementary Algebra

Question

Explain why \(2 x^{2} y\) and \(8 x y^{2}\) are not like terms.

Step-by-Step Solution

Answer

They are not like terms because the variables' exponents differ.

Step 1: Identify the terms

The terms given are \(2x^{2}y\) and \(8xy^{2}\).

Step 2: Break down each term into its factors

Break down the terms into their individual factors. \(2x^{2}y\) can be broken down into \(2 \cdot \ x^{2} \cdot \ y\). \(8xy^{2}\) can be broken down into \(8 \cdot \ x \cdot \ y^{2}\).

Step 3: Analyze the variables and their exponents

For the terms to be like terms, the variables and their exponents must match. In \(2x^{2}y\), the exponents for \(x\) and \(y\) are 2 and 1, respectively. In \(8xy^{2}\), the exponents for \(x\) and \(y\) are 1 and 2, respectively.

Step 4: Compare the terms

Compare the variables and their exponents from both terms. \(2x^{2}y\) has variables \(x^{2}\) and \(y\), while \(8xy^{2}\) has variables \(x\) and \(y^{2}\).

Step 5: Conclude they are not like terms

Since the variables' exponents do not match (one has \(x^{2}\) and the other has \(y^{2}\)), \(2x^{2}y\) and \(8xy^{2}\) are not like terms.

Key Concepts

Identifying Terms

To understand why certain algebraic expressions are not like terms, we first need to identify the terms themselves. In this case, the terms are: \(2x^{2}y\) and \(8xy^{2}\).
A term in algebra is a single mathematical expression. It could be a number, a variable, or several variables that are multiplied together. When looking at \(2x^{2}y\), the term consists of the coefficient 2 and the variables \(x^{2}\) and \(y\).
Similarly, for \(8xy^{2}\), the term consists of the coefficient 8 and the variables \(x\) and \(y^{2}\).
By breaking down these terms, we can better understand the components involved and prepare for further comparison.

Comparing Exponents

The next step is to compare the exponents of each variable in both terms. Exponents tell us how many times a variable is multiplied by itself. In the term \(2x^{2}y\), the exponent of \(x\) is 2, and the exponent of \(y\) is 1 (since any variable without an explicit exponent has an exponent of 1).
For the term \(8xy^{2}\), the situation is different. The exponent of \(x\) is 1, and the exponent of \(y\) is 2.
For terms to be considered like terms in algebra, each corresponding variable must have the same exponent in both terms. Since the exponents of \(x\) and \(y\) differ between \(2x^{2}y\) and \(8xy^{2}\), these terms are not like terms.
It's crucial to always check the exponents when comparing algebraic terms to determine if they are like terms.

Variables in Algebra

Variables play a significant role in algebra. They represent unknown values and are usually denoted with letters such as \(x\), \(y\), and \(z\). When dealing with expressions like \(2x^{2}y\) and \(8xy^{2}\), understanding the variables and their role is key to grasping why these terms are different.
Variables in algebra can have different powers or exponents and can be multiplied together with constants (known as coefficients).
For example:

• In \(2x^{2}y\), the variables are \(x\) and \(y\), with exponents of 2 and 1, respectively.
• In \(8xy^{2}\), the variables remain \(x\) and \(y\), but their exponents are 1 and 2, respectively.

The disparity in the exponents confirms that these terms are not like terms, as like terms must have identical variables raised to the same powers.
Recognizing the importance of variables and their exponents helps in successfully managing and simplifying algebraic expressions.