Problem 14 - Elementary and Intermediate Algebra

Question

Identify the base and the exponent in each expression. A. \(x^{5}\) B. \(-x^{5}\) C. \((-x)^{5}\)

Step-by-Step Solution

Answer

A: Base is \(x\), Exponent is \(5\). B: Base is \(x\), Exponent is \(5\). C: Base is \(-x\), Exponent is \(5\).

Step 1: Identify the base in expression A

In the expression \(x^5\), the base is the number or variable that is being raised to a power. For part A, the base is \(x\).

Step 2: Identify the exponent in expression A

The exponent in the expression \(x^5\) is the number that indicates how many times the base is multiplied by itself. For part A, the exponent is \(5\).

Step 3: Identify the base in expression B

In the expression \(-x^5\), the base is again \(x\), because the '-' sign is not included in the base when it's written outside the parentheses.

Step 4: Identify the exponent in expression B

The exponent in the expression \(-x^5\) is \(5\), as the raising power only concerns the variable \(x\).

Step 5: Identify the base in expression C

In the expression \((-x)^5\), the base is \(-x\). The parentheses indicate that the negative sign is included as part of the base.

Step 6: Identify the exponent in expression C

The exponent in the expression \((-x)^5\) is \(5\), as it applies to the entire base of \(-x\) inside the parentheses.

Key Concepts

Base and Exponent

The base and exponent are fundamental components in the world of mathematics, especially when dealing with powers.

• Base: The base is the number or variable that is repeated in multiplication. Think of it as the starting point or foundation. For example, in the expression \( x^5 \), the base is \( x \). This means \( x \) is the number you multiply by itself.
• Exponent: The exponent indicates how many times the base is multiplied by itself. It is usually written as a superscript number to the right of the base. So, in \( x^5 \), the exponent is \( 5 \). This guides us to multiply \( x \) five times: \( x \times x \times x \times x \times x \).

Understanding which number or variable serves as a base and which one acts as an exponent is crucial when solving algebraic problems.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations such as addition, subtraction, multiplication, and division. For instance, in the expression \( x^5 \), it's a simple algebraic expression with a single variable, \( x \), raised to a power. Even more complex expressions, like \( (-x)^5 \), still classify as algebraic expressions. Here, the negative sign is crucial, particularly when enclosed in parentheses.Some points to note about algebraic expressions include:

• Variables: Symbols that represent numbers, such as \( x \) in \( x^5 \).
• Coefficients: Numbers written before variables to signify multiplication. For example, in \(-2x^5\), \(-2\) is a coefficient.
• Operators: Symbols like \(+\), \(-\), \(\times\), and \(\div\) that denote operations on components.

Algebraic expressions form the core of equation-solving and help represent real-world situations mathematically.

Negative Sign in Exponents

The negative sign in exponents can appear in different contexts and is often a source of confusion for students. It’s important to distinguish how and where it's applied as it can change the meaning of the expression.

• Without Parentheses: In expressions like \(-x^5\), the negative sign is not part of the base. Here, the base remains \(x\), and the negative sign only affects the result after calculation. This expression represents \(-1 \times x^5\).
• With Parentheses: In expressions such as \((-x)^5\), the negative sign is part of the base. This means \(-x\) is multiplied by itself five times, resulting in a negative product (since there are five occurrences of \(-x\)).

This distinction is key in algebraic operations since the location of the parentheses alters whether the negative sign impacts the base or the entire result. Always be careful with parentheses in algebraic expressions as they can significantly affect the outcome.