Problem 17 - Algebra 2

Question

Simplify each expression. \(\frac{-3 m n^{3}}{21 m^{2} n^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{n}{7m}\).

Step 1: Identify Common Factors in the Numerator and Denominator

The expression is \(\frac{-3 m n^{3}}{21 m^{2} n^{2}}\). Identify common factors in the numerator and the denominator. Both contain factors of \(m\) and \(n\), and both \(-3\) and \(21\) are divisible by \(3\).

Step 2: Factor Out the Common Terms

In the expression, \(-3\) can be factored out from the numerator, and \(21\) can be factored into \(3 \times 7\) in the denominator. The common factor of \(m\) is \(m^{1}\) in the numerator and \(m^{2}\) in the denominator, and for \(n\), it's \(n^{2}\) in both. The expression becomes: \[\frac{-1 \cdot 3 \cdot m^{1} \cdot n^{3}}{7 \cdot 3 \cdot m^{2} \cdot n^{2}}\].

Step 3: Cancel Out the Common Factors

Cancel the common factors from the numerator and denominator. The \(3\)s cancel out, one \(m\) from the numerator and denominator cancels leaving \(m\) in the denominator, and \(n^{2}\) cancels leaving \(n\) in the numerator. The simplified expression becomes: \[\frac{-n}{7m}\].

Step 4: Write the Simplified Expression

After canceling out the common factors, the final simplified expression is:\(-\frac{n}{7m}\).

Key Concepts

Factoring

When simplifying rational expressions, one of the first and most crucial steps is factoring the expression into its individual components. Factoring is breaking down numbers or expressions into a product of their simpler factors.
For the expression \( \frac{-3 m n^{3}}{21 m^{2} n^{2}} \), let's look at each piece:

• Numerator: \(-3 m n^{3}\) can be seen as the product of \(-3\), \(m^1\), and \(n^{3}\).
• Denominator: \(21 m^{2} n^{2}\) can be expressed as a product of \(3 \times 7\), \(m^2\), and \(n^{2}\).

Both the numerator and the denominator of our rational expression contain numerical coefficients and variable parts. We factored the numerical component and variables separately to find commonality, making it easier to simplify the expression in later steps.

Canceling Common Factors

Once we've factored both the numerator and the denominator, the next step is to cancel out common factors. This means simplifying the expression by removing factors that appear in both the numerator and the denominator.
Let's check the expression:\[\frac{-1 \cdot 3 \cdot m^{1} \cdot n^{3}}{7 \cdot 3 \cdot m^{2} \cdot n^{2}}\]

• First, notice the \(3\) in both the numerator and the denominator; it cancels out.
• The variable \(m^1\) in the numerator and \(m^2\) in the denominator means that when we cancel the \(m\) in the numerator, one \(m\) remains in the denominator.
• The \(n^{2}\) cancels out completely from both, leaving \(n\) in the numerator.

Canceling these factors reduces complexity and helps us to see the simplest form of the expression.

Numerator and Denominator

Understanding the role of the numerator and the denominator is essential in simplifying rational expressions. In a fraction, the term above the line is the numerator, and the term below is the denominator.
Here's what happens in our expression:

• The numerator was initially \(-3mn^3\), which after factoring and canceling, simplified to \(-n\).
• The denominator started as \(21m^2n^2\) and after simplifying became \(7m\).

Both the numerator and the denominator hold factors that interrelate, allowing simplification. By carefully understanding how these two parts can be broken down and simplified, we obtain \(-\frac{n}{7m}\), the simplest form of the original problem. This comprehension of how to manipulate and reduce the parts is fundamental in algebra and will help simplify complex rational expressions efficiently.