Problem 3 - Algebra Form and Function

Question

Which of the expressions are equivalent to monomials in \(x ?\) $$ -x \cdot x^{2} $$

Step-by-Step Solution

Answer

Answer: Yes, the expression \(-x \cdot x^{2}\) is equivalent to the monomial \(-x^{3}\) in \(x\).

Step 1: Identify the given expression

The given expression is: $$ -x \cdot x^{2} $$

Step 2: Simplify the expression

We can simplify the expression by combining the terms: $$ -x \cdot x^{2} = -x^{1} \cdot x^{2} $$ Using the property of exponents \((x^{a} \cdot x^{b} = x^{a+b})\), we can combine the two terms: $$ -x^{1} \cdot x^{2} = -x^{1+2} = -x^{3} $$

Step 3: Determine whether the expression is a monomial

Now that the expression is simplified, we need to check if it meets the criteria for a monomial: 1. There is only one term: Yes, there is only one term, \(-x^3\). 2. The variable has a non-negative integer exponent: The exponent of \(x\) in our expression is 3, which is a non-negative integer. Since our expression meets both conditions, we can conclude that the given expression is equivalent to a monomial in \(x\).

Key Concepts

Exponents

Exponents are a fundamental element in mathematics, often used to simplify calculations involving repeated multiplication of the same number. When you write an expression like \(x^2\), the expression indicates that the base \(x\) is multiplied by itself two times. This is what we call "raising \(x\) to the power of 2."

Exponents follow specific rules that help in simplifying mathematical expressions. The most common exponent rule is the product of powers rule. It states that when you multiply two powers with the same base, you add their exponents. For example, \(a^m \times a^n = a^{m+n}\). This is an essential rule to understand when working with expressions involving exponents, as it helps simplify and combine terms efficiently.

Simplifying expressions

To simplify an expression means to write it in the most compact or simplest form. This involves using rules of exponents, among other operations, to reduce or combine terms with the same base or factors. In our example, the expression given is

• \(-x \cdot x^2\)

In this case, you can simplify the expression by recognizing that they have a common base and applying the exponent rules.

The expression is first converted to \(-x^1 \cdot x^2\), recognizing that \(x\) alone is just \(x^1\). Simplification is achieved by combining the exponents, resulting in \(-x^{1+2} = -x^3\). This technique of simplifying by using exponent rules is foundational in algebra, making complex expressions more manageable and easier to work with.

Properties of exponents

Properties of exponents are rules that explain how to handle mathematical operations involving powers. These rules are crucial in simplifying expressions and solving equations effectively. Below are some key properties:

• **Product of Powers**: As seen in the exercise, multiplying two exponents with the same base results in adding the exponents: \(x^m \cdot x^n = x^{m+n}\).
• **Power of a Power**: When raising an exponent to another power, you multiply the exponents: \((x^m)^n = x^{m \cdot n}\).
• **Power of a Product**: Distribute the exponent to all factors: \((xy)^n = x^n y^n\).
• **Zero Exponent**: Any number raised to the power of zero is 1: \(x^0 = 1\), provided that \(x eq 0\).

Applying these properties correctly leads to easier expression handling, which confirms the efficiency and practicality of using exponents in mathematical problem-solving.