Problem 24 - College Algebra

Question

Simplify each exponential expression $$ x y^{-3} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the exponent to simplify

The term which needs to be simplified is \(y^{-3}\). The goal is to write this term in such a way that its exponent will be positive.

Step 2: Simplify the expression

Apply the rule of exponents \(a^{-n} = 1/a^n\) to simplify the term \(y^{-3}\). This will give \(1/y^3\).

Step 3: Present the final form of the expression

Reassemble the expression. Replace \(y^{-3}\) with \(1/y^3\) in the initial expression \(x y^{-3}\). This transforms it into \(x * 1/y^3\) or more simply \(x/y^3\).

Key Concepts

Exponent Rules

Understanding exponent rules is vital for algebraic manipulation, especially when simplifying exponential expressions. An exponent represents the number of times a base is multiplied by itself. For example, in the expression \(x^3\), \(x\) is the base and is multiplied 3 times: \(x \times x \times x\). There are several key rules that can be applied to exponents:

• Product of Powers: When multiplying two expressions with the same base, add their exponents. \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: When dividing two expressions with the same base, subtract the exponents. \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: When raising an expression to a power, multiply the exponents. \( (a^m)^n = a^{mn}\).
• Negative Exponents: An expression with a negative exponent represents the reciprocal of the expression with a positive exponent. \(a^{-n} = \frac{1}{a^n}\).
• Zero Exponents: Any nonzero base raised to the zero power is equal to one. \(a^0 = 1\).

By applying these rules, students can systematically approach and simplify expressions involving exponents, turning potentially complex expressions into simpler ones.

Negative Exponents

Negative exponents can be confusing at first, but they adhere to a specific rule that makes working with them straightforward. A negative exponent indicates that the base is on the wrong side of the fraction—simply put, you have to take the reciprocal of the base to make the exponent positive.

Turning Negative Exponents into Positive

Take the expression \(a^{-n}\). According to exponent rules, this is equivalent to \(\frac{1}{a^n}\), where you have moved the base \(a\) from the numerator to the denominator and removed the negative by making the exponent positive. This is exactly the step taken in the problem at hand. The negative exponent \(y^{-3}\) becomes \(\frac{1}{y^3}\). When simplified, any term with a negative exponent will become a portion of a fraction, with the base of the term migrating to the denominator to satisfy the necessary positive exponent. This simple transformation is a critical step in ensuring that expressions are simplified correctly.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operations (like addition, subtraction, multiplication, and division). Simplifying algebraic expressions is a crucial skill in mathematics.

Simplifying Expressions

When simplifying an algebraic expression, the goal is to condense the expression into its simplest form. This process can involve a variety of steps, including combining like terms, factoring, expanding expressions, and applying exponent rules.

In the given exercise, we simplified the expression \(x y^{-3}\) by addressing the negative exponent, a step that follows the exponent rules. In the final form \(x/y^3\), there are no more exponents to simplify, and the expression is in its simplest form. Simplifying algebraic expressions often involves an understanding of multiple concepts simultaneously. By breaking down the expression into simpler components or combining terms to reduce complexity, one can rewrite expressions in a way that is easier to work with or understand.