Problem 18 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{-30 x^{2} y^{2} z^{2}}{-35 x z^{3}}\)

Step-by-Step Solution

Answer

Simplified expression: \(-\frac{6xy^2}{7z}\).

Step 1: Identify Common Factors

To simplify the rational expression \(-\frac{30x^2y^2z^2}{35xz^3}\), start by identifying any common factors in the numerator and the denominator. The coefficients 30 and 35 can both be divided by 5.

Step 2: Factor Out the Greatest Common Divisor

Divide both the numerator and the denominator by their greatest common divisor. The common factor for 30 and 35 is 5, so divide them to get: \(-\frac{30x^2y^2z^2}{35xz^3} = \frac{-6x^2y^2z^2}{7xz^3}\).

Step 3: Cancel Out Common Variables

Next, look at the variables. In the numerator \(x^2\) and in the denominator \(x\), reduce \(x^2 / x = x\). Similarly, reduce \(z^2 / z^3 = 1/z\). This gives us: \(\frac{-6xy^2}{7z}\).

Step 4: Combine and Simplify the Expression

After cancelling out similar terms, there are no further like terms to simplify. So, the simplified expression remains \(\frac{-6xy^2}{7z}\).

Key Concepts

Greatest Common Divisor

The Greatest Common Divisor (GCD) is an essential tool in simplifying rational expressions. It helps to identify the largest factor that divides both the numerator and the denominator without leaving a remainder. In our example, the coefficients of the numerator and denominator are 30 and 35, respectively. To find the GCD of 30 and 35:

• List the factors of each number: 30 has factors 1, 2, 3, 5, 6, 10, 15, and 30, while 35 has factors 1, 5, 7, and 35.
• Identify the largest common factor: Here, 5 is the greatest number that appears in both lists.

Dividing both numbers by 5 simplifies the coefficients and helps in reducing the overall rational expression. In algebra, using the GCD is a straightforward way to make expressions easier to work with.

Cancelling Common Terms

Once the greatest common divisor has been used to simplify the coefficients, the next step is cancelling common terms, particularly variables that appear in both the numerator and denominator. In rational expressions with variables, look for terms that are identical but in different degrees in the numerator and the denominator, as these can often be cancelled or reduced.
In the example:

• Notice the term \( x^2 \) in the numerator and the term \( x \) in the denominator: cancelling these results in just \( x \) in the numerator.
• Look at the \( z \) terms: \( z^2 \) in the numerator divided by \( z^3 \) in the denominator simplifies to \( 1/z \).

By cancelling these common terms, the expression is simplified to a much more manageable form. It’s important to handle the variables carefully to avoid errors.

Variables in Algebra

Variables in algebra represent unknown quantities and can stand in for numbers in expressions and equations. When simplifying rational expressions, understanding how to manipulate variables is crucial.
Variables are affected by operations with exponents and can often be cancelled when in the form of fractions. In our example, variables \( x \), \( y \), and \( z \) take center stage:

• \( x^2 \) indicates \( x \times x \), so when divided by \( x \), one \( x \) is eliminated, leaving one \( x \) in the numerator.
• The variable \( y^2 \) has no corresponding \( y \) term in the denominator, hence it remains unchanged.
• \( z^2 \) in the numerator and \( z^3 \) in the denominator means we effectively cancel two \( z \) terms and are left with \( 1/z \).

Handling variables correctly ensures that the simplified form is accurate. Always pay attention to exponents when simplifying, as they dictate how many times a variable is multiplied by itself.