Problem 14 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{-14 x^{2} y^{3}}{63 x y^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{-2xy}{9}\).

Step 1: Identify Common Factors

The first step in simplifying the rational expression \(\frac{-14x^2y^3}{63xy^2}\) is to identify any common factors in the numerator and the denominator. Observe that both terms can be factored into constants and variables: - Numerator: \(-14x^2y^3\) can be written as \(-1 \cdot 14 \cdot x^2 \cdot y^3\).- Denominator: \(63xy^2\) can be written as \(63 \cdot x \cdot y^2\).

Step 2: Factor the Constants

Factor the constants in both the numerator and the denominator:- Numerator: \(-14\) can be factored as \(-1 \times 2 \times 7\).- Denominator: \(63\) can be factored as \(9 \times 7\) or \(3 \times 3 \times 7\). The number 7 is common in both, and can be cancelled out.

Step 3: Cancel Common Factors

Identify the common factors to cancel out from both the numerator and the denominator:- **Common Constants:** Both 14 from the numerator and 63 from the denominator have a common factor of 7.- **Variables:** 1. For \(x\): Cancel one \(x\) from the numerator and one \(x\) from the denominator as both contain \(x\). 2. For \(y\): Cancel \(y^2\) from the denominator with \(y^2\) in the numerator, leaving \(y\) in the numerator.So, cancel \(7, x\), and \(y^2\) from the originally given expression.

Step 4: Simplify the Expression

After canceling out the common factors:- You're left with \(-2x^1y^{1}\) in the numerator and \(9\) in the denominator.Hence, the simplified expression is \(\frac{-2xy}{9}\).

Key Concepts

Understanding Common Factors

When simplifying rational expressions, a crucial step involves identifying common factors in both the numerator and the denominator. A common factor is any factor that can divide both terms in the fraction without a remainder. To illustrate, let's look at our given expression: \[\frac{-14x^2y^3}{63xy^2}\] Here, both the numerator and the denominator can be broken down into

• Numerator: -1 \cdot 14 \cdot x^2 \cdot y^3
• Denominator: 63 \cdot x \cdot y^2

The task is to identify shared elements between these terms. Such elements are called common factors. Observing both the numerator and the denominator, the terms 7, \( x \), and \( y^2 \) appear in both, which are the common factors here. Recognizing these factors allows you to significantly reduce the complexity of the expression.

The Art of Factoring

Factoring is a powerful algebraic technique used to break down numbers or expressions into their simpler parts—called factors. This simplification is vital in making expressions easier to work with. For our specific expression: \[\frac{-14x^2y^3}{63xy^2}\] Let's factor each part:

• Numerator: \(-14\) is \(-1 \times 2 \times 7\). For variables,\( x^2 \) is \( x \times x \) and \( y^3 \) is \( y \times y \times y \).
• Denominator: 63 can be factored as \( 3 \times 3 \times 7 \). Variables \( x \) and \( y^2 \) are straightforward as \( x \) and \( y \times y \).

Factoring is all about rephrasing your terms into building blocks. Once factored out, it becomes easy to see which pieces of the numerator and denominator can be canceled. Factoring lays the foundation for effective simplification.

Canceling Terms Effectively

Once you've identified the common factors and factored both the numerator and the denominator, the next step is canceling these factors. Canceling refers to the process of removing these common factors from both parts of the fraction, thereby simplifying the expression. For example, in our given expression: \[\frac{-14x^2y^3}{63xy^2}\] The factors that we can cancel include:

• The number 7 from both 14 and 63.
• One instance of \( x \) from both \( x^2\) and \( x \).
• Two instances of \( y \) (i.e., \( y^2\)) from both \( y^3 \) and \( y^2 \).

Canceling these common elements reduces our expression to: \[\frac{-2x^1y^1}{9}\] Canceling is a critical step as it trims down the expression to its most simplified form. Remember: you can only cancel factors, not just terms randomly positioned in the expression.

Final Simplification Steps

Simplification is the process of making an algebraic expression as easy to read and as straightforward as possible. After factoring and canceling common terms, the final expression should be as reduced as it can get. Our result from the previous section was: \[\frac{-2xy}{9}\] In this expression, all possible simplifications have been made. Let's check:

• All common factors have been canceled correctly.
• No whole numbers or variables are shared between numerator and denominator.
• Expression is reduced to its simplest form.

The ultimate goal of simplification is to present the expression in a form that's as uncomplicated as possible. This not only helps with solving equations but also in understanding and communication. Mastering the process of simplification enriches your ability to tackle more complex algebraic expressions efficiently.