Problem 41 - Algebra 1

Question

List the terms of the expression. $$-(y)^{3}$$

Step-by-Step Solution

Answer

The expression \(-(y)^{3}\) has a single term, which is \(-(y)^{3}\) itself.

Step 1: Understand the Expression

The expression given is \(-(y)^{3}\). This is a single term because it is not joined by plus (+) or minus (-) to any other terms.

Step 2: Analyze the Sign

The sign of the term is negative because it is preceded by a ‘-’ (minus) sign. Hence, this makes it important to remember the term is negative.

Step 3: Analyze the Power

In terms of power, the term \(y\) is cubed, which is represented as \(y^{3}\). This is one of the fundamental parts of the term.

Key Concepts

Terms of an Expression

Let’s start by breaking down what we mean by terms of an expression. In algebra, an expression is made up of terms that are the building blocks of the mathematical phrase. These can include numbers, variables (like \( x \) or \( y \)), and the operations that connect them, such as addition or subtraction.

A term can stand alone, or it can be a part of a larger expression. For instance, in the expression \( 3x + 4y - 5 \), there are three terms: \( 3x \), \( 4y \), and \( -5 \). Each term consists of a coefficient (a number) and a variable part, except for \( -5 \), which is a constant term as it does not contain any variable.

In the context of the given problem, \( -(y)^{3} \) is considered a single term. There are no additional terms because it is not combined with any other numbers or variables through addition or subtraction. Understanding how terms work in an algebraic expression is crucial to mastering algebra and progressing to more complex problems.

Negative Exponents

Now let's clarify a common source of confusion - negative exponents. Negative exponents indicate that we are dealing with a reciprocal of a number or variable raised to a positive exponent. For instance, \( x^{-n} \) is equivalent to \( \frac{1}{x^{n}} \), where \( n \) is a positive integer.

However, it’s essential to note that a negative sign in front of an expression, such as \( -(y)^{3} \), is not the same as having a negative exponent. The negative in \( -(y)^{3} \) simply shows that the entire term is negative, rather than suggesting the reciprocal of a cubed term. Always pay close attention to where the negative sign is placed to understand the value and the meaning it implies in algebraic expressions.

Cubed Terms

Cubed terms involve raising a number or a variable to the third power, which is visually represented by the exponent of three, such as in \( a^{3} \). This means that you are multiplying the base (\( a \)) by itself three times: \( a \times a \times a \).

For example, if we have a term like \( 8^{3} \), it can be expanded to \( 8 \times 8 \times 8 \), which equals 512. Similarly, when we have a variable raised to the third power, like \( y^{3} \), it means that we are multiplying the variable \( y \) by itself twice more: \( y \times y \times y \).

In our exercise, we have \( -(y)^{3} \), meaning that whatever the value of \( y \) is, it will be cubed, and then made negative because of the minus sign in front. Understanding cubed terms is important as it helps us manipulate and simplify expressions effectively in algebra.