Problem 107 - College Algebra

Question

In Exercises 107–114, simplify each exponential expression.Assume that variables represent nonzero real numbers. $$\frac{\left(x^{-2} y\right)^{-3}}{\left(x^{2} y^{-1}\right)^{3}}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is 1.

Step 1: Distribute the Exponent

In the first step, distribute the negative exponent to both variables in the numerator and simplify each factor separately. The same process should be applied to the denominator. Use the rule that \(a^{-n} = \frac{1}{a^n}\): \[\frac{(x^{-2} y)^{-3}}{(x^{2} y^{-1})^{3}} = \frac{x^{6} y^{-3}}{x^{6} y^{-3}}\]

Step 2: Cancel common factors

Identify any common factors in the numerator and the denominator, and cancel them out to simplify the expression. Each instance of \(x^6\) and \(y^{-3}\) will cancel each other since they exist both in the numerator and the denominator. Therefore: \[ \frac{x^{6} y^{-3}}{x^{6} y^{-3}} = 1\]

Key Concepts

Simplifying Expressions

Simplifying expressions often involves rewriting them in a form that is easier to understand or compute. When dealing with exponential expressions like \[ \frac{(x^{-2} y)^{-3}}{(x^{2} y^{-1})^{3}} \], it is crucial to follow a systematic approach across multiple steps.
Start by distributing any exponents across the terms in parentheses. You apply the exponent to each base inside the parentheses, which simplifies the structure. In the example, \((x^{-2} y)^{-3}\) and \((x^{2} y^{-1})^{3}\) are expanded by applying the outer powers to each inner component.
This method ensures each term becomes a distinct factor with new exponents. By simplifying the expression, comparable factors can easily be compared and reduced. This step-by-step approach bolsters understanding and aids in achieving the simplest form.

Negative Exponents

Negative exponents indicate the reciprocal of that base raised to the opposite positive exponent. It's like reversing the direction of multiplication to division. For instance, \( x^{-2} \) is the same as \( \frac{1}{x^2} \).
When you encounter negative exponents, convert them by taking the reciprocal of the base and changing the sign of the exponent to positive.
Let's take a negative example from the equation: \((x^{-2} y)^{-3}\): here, each negative exponent when distributed turns it into a positive by converting the base, resulting in each term becoming its reciprocal raised to a now positive power.

• \(x^{-2}\) becomes \(x^2\) when taken to a negative power of -3, simplifying to \(x^6\).
• \(y\) stays \(y^{-3}\), reflecting the introduction of power.

Converting negative exponents in this way is key to simplifying the expression.

Algebraic Manipulation

Algebraic manipulation involves reorganizing and simplifying expressions by recognizing equivalent expressions and common factors. For a complex fraction like \( \frac{x^6 y^{-3}}{x^6 y^{-3}} \), the goal is to reduce it through strategic cancelation.
Identify identical factors in the numerator and denominator to employ straightforward cancelations. In this example, \(x^6\) and \(y^{-3}\) are common and thus eliminated, simplifying the entire fraction to 1.
This illustrative step highlights how breaking down and rearranging terms can reveal hidden simplicity in an expression.

• Recognize like terms.
• Cancel identical terms from numerator and denominator.

Through such manipulation, layers of complexity are peeled away, achieving a clean, simplified result.