Problem 19 - Algebra 1: Concepts and Skills

Question

Simplify the expression. If not possible, write already in simplest form. $$\frac{14 x^{2}}{50 x^{4}}$$

Step-by-Step Solution

Answer

\(\frac{7}{25 x^{2}}\)

Step 1: Identify Common Factors

The numerator and the denominator of the given expression \(\frac{14 x^{2}}{50 x^{4}}\) have a common factor \(x^{2}\). So we divide both the numerator and the denominator by \(x^{2}\). The numerical values 14 and 50 have a common factor of 2, which can also be divided from both the numerator and the denominator.

Step 2: Divide through by the Common Factors

Divide the numerator and the denominator by \(x^{2}\), this gives: \(\frac{14}{50 x^{2}}\). Now, divide 14 and 50 by their common factor 2, this yields: \(\frac{7}{25 x^{2}}\).

Step 3: Check If Further Simplification Is Possible

Since there are no more common factors in the numerator and the denominator, the expression \(\frac{7}{25 x^{2}}\) is already in its simplest form.

Key Concepts

Common Factors

In algebra, finding the common factors of terms in a fraction is a key step in simplifying them. Common factors are values that are shared by two or more numbers or terms in an expression.
For example, in the expression \(\frac{14 x^{2}}{50 x^{4}}\), the common factors between both parts of the fraction are critical.

• Identify elements that both the numerator and denominator share. In this case, it's useful to observe both the numeric part and the algebraic part separately.
• Here, the numbers 14 and 50 have a common factor of 2. Additionally, the terms \(x^{2}\) in the numerator and \(x^{4}\) in the denominator share a power of \(x^{2}\).
• Dividing both the numerator and the denominator by these common factors will simplify the fraction.
  

Spotting common factors is your starting point for simplification.

Numerator and Denominator

The numerator and denominator are the top and bottom parts of a fraction, respectively. They are fundamental in understanding how fractions work. For the fraction \(\frac{14 x^{2}}{50 x^{4}}\):

• The **numerator** is 14\(x^{2}\). This is the part we are simplifying by identifying its common factors with the denominator.
• The **denominator** is 50\(x^{4}\). It forms the bottom framework of our fraction and in our simplification process, is treated similarly to the numerator.
• Simplifying a fraction means dividing both parts by their common factors, without altering the value of the expression.

Focusing on the numerator and denominator separately helps clarify how to approach simplification.

Simplest Form

Achieving the simplest form of an algebraic fraction involves reducing it until no common factors remain. For the expression \(\frac{14 x^{2}}{50 x^{4}}\), you want to reach a state where further simplification isn't possible.

How to Determine the Simplest Form

• After dividing by known common factors (e.g., \(2\) and \(x^{2}\) here), re-evaluate the result. The expression becomes \(\frac{7}{25 x^{2}}\).
• Check for additional common factors by inspecting both the numerator and denominator again. Neither 7 nor 25 have common factors, nor do \(x^{0}\) and \(x^{2}\), except for 1.
• Conclude that the expression is in its simplest form when all possible common factors are accounted for.
  

Using common factors correctly ensures that your expression is as simple as possible.