Problem 23 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$x^{-2} y$$

Step-by-Step Solution

Answer

The simplified form of the exponential expression \(x^{-2} y\) is \(\frac{y}{x^2}\).

Step 1: Convert Negative Exponent to Positive

The general rule for converting a negative exponent to a positive exponent is to take the reciprocal of the base. So we need to rewrite the \(x^{-2}\) as \(\frac{1}{x^2}\).

Step 2: Combine the revised expression with 'y'

Combine the simplified term \(\frac{1}{x^2}\) with the 'y' from the initial expression resulting to \(\frac{y}{x^2}\).

Key Concepts

Negative Exponent Rule

When dealing with exponential expressions, a negative exponent can often cause confusion, but the rule for handling them is quite straightforward. The negative exponent rule states that a term with a negative exponent can be converted into a positive exponent by taking the reciprocal of its base. For example, if we have the term \( x^{-2} \), by applying this rule, we rewrite it as \( \frac{1}{x^2} \). This transformation is crucial as it simplifies the expression and allows for further algebraic manipulation.

Understanding this rule is essential when simplifying complex algebraic expressions, as it often appears in calculus, physics, and other mathematical contexts. To apply this rule effectively, remember that a negative exponent 'flips' the position of the base: if it was in the numerator, it goes to the denominator, and vice versa.

Reciprocal of a Base

The concept of the reciprocal is fundamental in mathematics. The reciprocal of a base is simply the inverse of a number or expression. If \( x \) is a nonzero number, then the reciprocal of \( x \) is \( \frac{1}{x} \). In terms of exponents, the reciprocal also applies to the base of a power. For example, the reciprocal of \( x^2 \) is \( \frac{1}{x^2} \), as shown in our exercise.

It's important when working with fractions and division within algebraic expressions. When dealing with exponents, taking the reciprocal is synonymous with switching the sign of the exponent from positive to negative, or vice versa.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \( x \) or \( y \)), and operations (such as add, subtract, multiply, and divide). The beauty of algebraic expressions lies in their ability to represent real-world situations abstractly and their versatility in being manipulated according to algebraic rules to simplify or solve equations.

In our exercise, \( x^{-2} y \) is an algebraic expression combining a variable with a negative exponent and another variable. Simplifying expressions like these often involves applying exponent rules, factoring, combining like terms, or other algebraic techniques to make the expression more understandable or ready for solving an equation.

Exponential Notation

Exponential notation is a convenient way to express repeated multiplication of the same factor. In this notation, a number called the base is raised to an exponent, which tells us how many times to multiply the base by itself. For instance, \( x^3 \) means \( x \) multiplied by itself three times: \( x \times x \times x \).

This notation becomes especially powerful when combined with exponent rules, such as the negative exponent rule, to simplify expressions. Exponential notation is fundamental in various areas of mathematics and science, as it can represent very large or very small numbers succinctly and is central to the laws of exponents, which govern how to manipulate powers of the same base.