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Simplifying fractions is a process that reduces fractions to their simplest form. A fraction is simplified when the numerator and the denominator are reduced to their smallest possible values, which cannot be divided by any common factor other than 1.

The key to simplifying is identifying any common factors shared by both the numerator and denominator. The ultimate aim is to make calculations easier and the fraction more understandable. For example, take the fraction \( \frac{21}{168} \). The numerator is 21 and the denominator is 168. You find a common factor, which in this case is 21, and divide both by this number.

After reducing by 21, we get \( \frac{1}{8} \), which means 21 goes into 21 once and into 168 eight times. This process of division by the greatest common factor ensures the fraction is as simple as possible.

The Greatest Common Factor (GCF) is the largest number that divides two or more numbers without leaving a remainder. It is an essential tool for simplifying fractions.

To find the GCF of two numbers, list out all the factors of each number, then identify the largest factor that appears in both lists. For example, to find the GCF of 21 and 168, list the factors:

• Factors of 21: 1, 3, 7, 21
• Factors of 168: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168

The greatest common factor is 21, as it is the largest number present in both lists.

Applying the GCF helps simplify not only numerical fractions but also algebraic expressions, by reducing complex polynomial fractions to simpler forms.

Polynomial multiplication involves multiplying two or more polynomials together. Here, each term in one polynomial is multiplied by every term in the other polynomial. The process often includes combining like terms to reach a simplified expression.

Let's break down the multiplication of the numerators from the original exercise: \(3m^2\) and \(7mn\).

First, multiply the coefficients (the numbers in front of the variables), which are 3 and 7. This gives us 21. Next, multiply the variables by combining their exponents due to the rule \(x^a \cdot x^b = x^{a+b}\) when the bases are the same. So for \(m^2\) and \(m\), add the exponents to get \(m^3\). Don't forget that \(n\) is already a separate variable. As a result, the product is \(21m^3n\).

This systematic approach helps manage the multiplication of more complex polynomial expressions, ensuring accuracy and ease in dealing with algebraic problems.