Problem 16 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{60 x^{3} z}{-64 x y z^{2}}\)

Step-by-Step Solution

Answer

\(\frac{15x^2}{-16yz}\).

Step 1: Identify Common Factors

First, identify the greatest common factors in the numerator and the denominator of the rational expression. Look for numerical factors as well as variables that are present in both the numerator and denominator. Here, 60 and -64 share a common factor of 4. For the variables, we have: \(x^3\) and \(x\), and \(z\) and \(z^2\). These can be reduced as well.

Step 2: Simplify the Numerical Coefficients

Divide the numerical coefficient in the numerator by its common factor with the denominator. Simplify:\[\frac{60}{-64} = \frac{15}{-16}\] because 60 divided by 4 is 15 and -64 divided by 4 is -16.

Step 3: Simplify the Variable Factors

Simplify the variable expressions by canceling out the common variables. Since \(x^3\) and \(x\) both share \(x\) as a factor, that becomes \(x^{2}\). For \(z\) and \(z^2\), divide \(z\) by \(z\) to leave \(z\) in the denominator. The expression becomes:\[ \frac{x^{2}}{yz} \].

Step 4: Assemble the Simplified Expression

Combine the simplified numerical and variable expressions together. The reduced rational expression is:\[\frac{15x^2}{-16yz}\].

Key Concepts

Common Factors

When simplifying rational expressions, the first step is to look for common factors in both the numerator and the denominator.
For our expression, \(\frac{60 x^{3} z}{-64 x y z^{2}}\), identify factors present in both the top and bottom of the fraction.
This includes both numerical factors and variables:

• Numerical Factors: Both 60 and -64 can be divided by 4, as it is their greatest common factor.
• Variables: In terms of variables, we have \(x^3\) and \(x\), and \(z\) and \(z^2\). These variables also share common elements that can be reduced.

Understanding these common factors is essential, as it helps streamline expressions to their simplest form.

Numerical Coefficients

Numerical coefficients are the number parts of terms in an expression.
In our rational expression, \(\frac{60 x^{3} z}{-64 x y z^{2}}\), the numerical coefficients are 60 and -64.
Once common numerical factors are identified, simplify them by division:

• Divide 60 by 4 to obtain 15.
• Similarly, divide -64 by 4 to get -16.

This will refine the numerical part of your expression to \(\frac{15}{-16}\).
This step is crucial in reducing complex fractions into a more manageable and simpler form.

Variable Cancellation

Variable cancellation involves reducing expressions by eliminating common variable factors found in both numerator and denominator.
In the expression \(\frac{60 x^{3} z}{-64 x y z^{2}}\), you first identify the common variable factors:

• Both \(x^3\) in the numerator and \(x\) in the denominator share at least \(x\), resulting in \(x^2\) remaining in the numerator.
• Similarly, \(z\) in the numerator and \(z^2\) in the denominator share \(z\), leaving a single \(z\) in the denominator.

This simplifies the expression to \(\frac{x^{2}}{yz}\) by effectively canceling out these common factors. Such cancellation makes complex terms easier to evaluate and work with.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest number or variable that divides two or more numbers or terms without leaving a remainder.
Finding GCF is a powerful tool in simplification:

• For numbers like 60 and -64, the GCF is 4. This helps to simplify \(\frac{60}{-64}\) to \(\frac{15}{-16}\).
• For variables \(x^3\) and \(x\), \(x\) is the common variable factor.
• For \(z\) and \(z^2\), \(z\) is uncovered as the GCF.

By using the GCF, you ensure that you are simplifying the expression as much as possible, which sets the platform for solving more complex algebraic problems with ease.