91Ó°ÊÓ

In the world of rational expressions, denominators play a crucial role. The denominator is the number or variable located at the bottom of a fraction. It tells us how many parts the whole is divided into. For example, in the fraction \( \frac{1}{2} \), the number '2' is the denominator. It indicates that the whole is divided into two equal parts.

When dealing with algebraic expressions, the denominator can include variables. In the exercise provided, the denominator is \(m-9\) in the original expression and \(m(m-9)\) in the rewritten version. The goal is often to rewrite the rational expression with a specified denominator, to make equations more comparable or to prepare them for additional operations.

When you change a denominator, it's vital to adjust the numerator accordingly, to ensure the expression remains equivalent.

Multiplying fractions involves multiplying the numerators together and multiplying the denominators together. This technique is utilized when finding a common denominator or when simplifying expressions.

In the original solution, the rational expression \( \frac{5}{m-9} \) needed to change to have a denominator of \( m(m-9) \). To achieve this, you multiply the numerator and the denominator by the missing factor, which is not present in the original denominator—in this exercise, that's 'm'.

This is because multiplying by \( \frac{m}{m} \) is essentially multiplying by 1, so the value of the expression doesn’t change. This preserves the fraction's equivalence while transforming its form to match the desired denominator.

Simplifying expressions is about rewriting them in a more understandable or usable form, without changing their value. For rational expressions, simplifying often involves multiplying, factoring, or canceling terms in the numerator and the denominator.

In this exercise, once the multiplication by the factor 'm' was performed on \( \frac{5}{m-9} \), we reached the expression \( \frac{5m}{m(m-9)} \).

This expression is simplified in the sense that it has arrived at the form with the indicated denominator, but no further simplification by canceling is needed as no common factors remain across the numerator and the denominator. Simplification ensures that expressions are as concise and manageable as possible.

Algebra is a branch of mathematics that uses symbols, usually letters, to represent numbers or quantities in equations and formulas. It allows us to solve problems involving unknown or variable quantities.

In the context of rational expressions, algebraic techniques help manipulate and rearrange fractions to meet certain conditions—like rewriting an expression with a particular denominator. Understanding the basic rules of algebra, such as maintaining balance by performing the same operations on both sides of an equation (or both the numerator and denominator of a fraction), is key.

This exercise demonstrates how algebra is utilized not just to solve equations but to transform them, making it a practical tool for simplifying, solving, and understanding mathematical expressions.