Problem 9 - Elementary and Intermediate Algebra

Question

Fill in the blanks. To _____ a rational expression, we multiply it by a form of 1. $$\text { For example, } \frac{2}{n^{2}} \cdot \frac{8}{8}=\frac{16}{8 n^{2}}$$

Step-by-Step Solution

Answer

To simplify a rational expression, we multiply it by a form of 1.

Step 1: Identify the Objective

To simplify a rational expression, the goal is often to either simplify the numerator and denominator to their simplest form or rationalize the expression. However, it can involve manipulating the expression in a way that keeps its value unchanged.

Step 2: Understanding the Example

Analyze the given example: \(\frac{2}{n^{2}} \cdot \frac{8}{8} = \frac{16}{8 n^{2}}\). Here, the expression \(\frac{8}{8}\) is equivalent to 1 and is used to change the form of \(\frac{2}{n^{2}}\) without changing its value.

Step 3: Identify what is Multiplied by

In the example, we multiply \(\frac{2}{n^{2}}\) by \(\frac{8}{8}\), which is essentially multiplying by 1. This ensures the expression doesn't change its actual value but can help in simplification or reformation of the expression.

Step 4: Determining the Missing Word

Since \(\frac{8}{8} = 1\), the term used in the blank is the phrase that means keeping the rational expression's value constant while possibly changing its form. This term is 'simplify' or 'reform', but the process specifically depicted is multiplying by a 'form of 1'.

Key Concepts

Multiplying by Form of One

A clever technique we use to simplify rational expressions is multiplying by 1. But instead of the number one, we use a "form of one." This means using fractions like \( \frac{8}{8} \), \( \frac{x}{x} \), or any expression where the numerator and the denominator are the same and nonzero.
This strategy is powerful because multiplying by 1 maintains the value of the original expression while allowing us to modify its form.

• For example, with \( \frac{2}{n^2} \cdot \frac{8}{8} = \frac{16}{8n^2} \), we use \( \frac{8}{8} \) as our form of one.
• This multiplication doesn't change the rational expression's value. Still, it can help us either simplify it or set it up for further operations.

Multiplying by a form of one helps when we're looking to rewrite expressions for easier computation or comparing different expressions.

Rational Expressions

Rational expressions are essentially fractions where the numerator and the denominator are polynomials. They are a critical component in algebra and higher mathematics.

These expressions act much like standard fractions, where you can perform similar operations such as addition, subtraction, multiplication, and division.

• For instance, a rational expression could look like \( \frac{2}{n^2} \).
• It's crucial to ensure that the denominator never equals zero because dividing by zero is undefined.

Understanding rational expressions plays a critical role in managing algebraic equations, inequalities, and functions. They form the foundation for more advanced topics in mathematics.

Simplification Techniques

Simplification techniques are essential in mathematics because they help make expressions more manageable. For rational expressions, simplification possible involves reducing the expression to its simplest form.

The process often involves factoring polynomials in the numerator and denominator and canceling common factors.

• For example, if you have \( \frac{2}{8n^2} \), you could simplify it by factoring it to \( \frac{1}{4n^2} \).
• Using methods like multiplying by a form of one, you can adjust the expression's form to assist with factoring or aligning like denominators.

Simplifying expressions is not just about "making them look nicer"; it's about paving the way for easier computations and better understanding in mathematical problem-solving.