Problem 63 - Elementary and Intermediate Algebra

Question

Build each rational expression into an equivalent expression with the given denominator. \(\frac{8}{x} ; x^{2} y\)

Step-by-Step Solution

Answer

The equivalent expression is \(\frac{8xy}{x^2 y}\).

Step 1: Understand the Requirement

We need to rewrite the rational expression \(\frac{8}{x}\) so that it has the new denominator \(x^2 y\). This involves multiplying both the numerator and denominator by the same term to achieve the desired denominator.

Step 2: Identify the Missing Factors

The current denominator is \(x\), and the desired denominator is \(x^2 y\). The missing factors in the current denominator to reach \(x^2 y\) are \(xy\).

Step 3: Multiply Numerator and Denominator

Multiply both the numerator and the denominator of \(\frac{8}{x}\) by the missing factors \(xy\):\[\frac{8}{x} \times \frac{xy}{xy} = \frac{8xy}{x^2 y}\]

Step 4: Simplify the Expression

The expression \(\frac{8xy}{x^2 y}\) is already simplified as the desired denominator is achieved, and both \(xy\) are correctly multiplied.

Key Concepts

Equivalent Expressions

When dealing with rational expressions, an important concept is equivalent expressions. These are expressions that look different yet have the same value. The process of finding an equivalent expression involves transforming the original expression without changing its value.
For rational expressions, like \(\frac{8}{x}\), we want to rewrite them with a new denominator while keeping the expression equivalent to itself. This means multiplying both the numerator and the denominator by the same factor. By doing this, you are essentially multiplying the original expression by 1 (since any number over itself is 1), which doesn’t change the value of the expression.
Creating equivalent expressions is a useful algebraic tool, often applied to simplify fractions, solve equations, and combine rational expressions with different denominators.

Denominators

The denominator is a fundamental part of any rational expression. It is the term below the fraction line that indicates what the whole is divided by. Understanding and working with denominators is crucial for tasks like simplification and finding common denominators among different fractions.
In the exercise given, the original denominator is \(x\). The goal is to adjust this to the new, more complex denominator \(x^2 y\). To achieve this, we identify what factors are missing from the original denominator to reach the desired form. This involves a comparison between the two denominators, determining the factors needed (in this case, \(xy\)) to transform the original into the new denominator.
This is a key step in making sure all expressions have a common baseline, allowing for further operations like addition or subtraction of rational expressions to proceed smoothly.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions using a set of mathematical rules. This skill is essential in transforming and simplifying rational expressions.
In the case of our exercise, algebraic manipulation is used to successfully rewrite \(\frac{8}{x}\). First, by identifying the missing factors \(xy\). Then, by multiplying both the numerator and denominator by this factor, we effectively modified the fraction to \(\frac{8xy}{x^2 y}\). This manipulation ensures that the expression keeps its value, while adopting the required form.
Proper algebraic manipulation allows one to maintain balance in equations and solve for variables. It’s a creative process and with practice, it becomes second nature to find and apply the appropriate steps, making mathematical problems easier and more intuitive to handle.