Problem 24 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$x y^{-3}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(x y^{-3}\) is \(x / y^3\).

Step 1: Identify the exponent laws applicable

In the expression \(x y^{-3}\), we find an exponent of \(-3\) on the variable \(y\). This indicates the usage of the law \(a^{-n} = 1 / a^n\). Here \(a\) refers to \(y\) and \(n\) refers to 3.

Step 2: Apply Negative Exponent Rule

Apply the above mentioned law \(a^{-n} = 1 / a^n\) where \(a = y\) and \(n = 3\). This law will turn the expression into \(x / y^3\).

Key Concepts

Exponent Laws

Exponent laws, also known as the laws of exponents, are a set of rules that govern the operations on expressions with exponents. They are essential to simplify algebraic expressions that involve exponents. One of the fundamental laws is the product of powers rule, which states that when you multiply two terms with the same base, you can add their exponents:

For example, \( a^m \times a^n = a^{m+n} \).

Another important rule is the power of a power rule, which tells us that when you raise an exponent to another exponent, you multiply the exponents:

For example, \( (a^m)^n = a^{m \times n} \).

These laws make complex multiplication and division of exponential terms manageable and form the foundation for simplifying expressions involving powers.

Negative Exponent Rule

The negative exponent rule is a specific case within the exponent laws, which addresses how to handle negative exponents in algebraic expressions. This rule states that a term with a negative exponent can be transformed into its reciprocal with a positive exponent. Put simply:

\( a^{-n} = 1 / a^n \)

This allows us to rewrite expressions with negative exponents as fractions with positive exponents in the denominator. For instance, in the exercise where we have \( y^{-3} \), applying the negative exponent rule changes this to \( 1/y^3 \). Remember, this rule helps avoid negative exponents by converting them into a more familiar form, maintaining the simplicity and readability of the expression.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations such as addition, subtraction, multiplication, and division. They don't contain an equals sign, differentiating them from equations. Simplifying an algebraic expression means to reduce it to the simplest form possible while conveying the same information. This often involves combining like terms, factoring, expanding expressions, and applying exponent laws. For example, in our exercise, the expression \( x y^{-3} \) once simplified by applying the negative exponent rule becomes \( x / y^3 \). Simplifying algebraic expressions is a fundamental skill in algebra and helps in solving more complex problems with ease and accuracy.