Problem 22 - Elementary and Intermediate Algebra

Question

Divide the monomials. See Example 1. $$ \frac{3 y^{3} z}{18 y z^{6}} $$

Step-by-Step Solution

Answer

\( \frac{y^2}{6z^5} \)

Step 1: Identify Common Factors

First, identify the common factors in the coefficients of the numerator and the denominator. In the expression \( \frac{3 y^{3} z}{18 y z^{6}} \), the coefficients are 3 and 18. The greatest common factor (GCF) of 3 and 18 is 3.

Step 2: Divide the Coefficients

Divide the coefficients of the numerator and the denominator by the GCF. For the given problem, divide 3 by 3 to get 1 and divide 18 by 3 to get 6. This simplifies the expression to: \( \frac{1 y^{3} z}{6 y z^{6}} \).

Step 3: Simplify the Variable y

Next, simplify the variable \( y \). In the numerator, we have \( y^3 \), and in the denominator, we have \( y \). Subtract the exponents: \( 3 - 1 = 2 \). This gives us \( y^2 \) in the numerator. The expression now becomes: \( \frac{y^2 z}{6 z^{6}} \).

Step 4: Simplify the Variable z

Now simplify the variable \( z \). In the numerator, we have \( z^1 \), and in the denominator, we have \( z^6 \). Subtract the exponents: \( 1 - 6 = -5 \). Since \( z^{-5} \) is equivalent to \( \frac{1}{z^5} \), the expression becomes: \( \frac{y^2}{6z^5} \).

Step 5: Write the Final Expression

Combine all the simplifications to write the final expression. The simplified result of the division is: \( \frac{y^2}{6z^5} \).

Key Concepts

Understanding the Greatest Common Factor (GCF)

The greatest common factor, often abbreviated as GCF, is a fundamental concept in algebra that helps simplify complex equations. It's all about finding the largest number that divides two or more numbers without leaving a remainder. This simplifies expressions and makes calculations more manageable.
For instance, in the expression \( \frac{3 y^{3} z}{18 y z^{6}} \), the coefficients are 3 and 18. Here, the GCF of 3 and 18 is 3. This means both numbers are divisible by 3 without leaving a remainder. To use the GCF in simplifying monomials:

• Identify the coefficients in both the numerator and the denominator.
• Determine the largest common factor they share.
• Divide both coefficients by this factor to simplify.

This method reduces the complexity of algebraic expressions and keeps calculations smooth and error-free.

Simplifying Exponents in Algebraic Expressions

When dealing with expressions that involve exponents, simplifying them can make the calculations straightforward. Simplification often involves the "power rule," one of the most powerful tools you can have.
The power rule states that when you divide terms with the same base, you subtract their exponents: \( a^m \div a^n = a^{m-n} \).
In our exercise, simplifying starts with addressing the \( y \) and \( z \) exponents:

• For \( y^3 \) divided by \( y^1 \): Subtract the exponent in the denominator from that in the numerator to get \( y^{3-1} = y^2 \).
• For \( z^1 \) divided by \( z^6 \): Subtract the exponents to get \( z^{1-6} = z^{-5} \). This result tells us to bring \( z^5 \) to the denominator.

Applying these simple steps with the subtraction of exponents dramatically simplifies our work, leading us immediately to simplified expressions.

Working with Rational Expressions

Rational expressions involve fractions where the numerator and the denominator are both polynomials. Dividing these expressions requires a combination of skills including finding the greatest common factor, simplifying exponents, and ensuring the result retains proper form.
When simplifying rational expressions such as \( \frac{3 y^{3} z}{18 y z^{6}} \), it is crucial to:

• Simplify each part starting with coefficients, using the GCF.
• Combine and simplify exponents using subtraction.
• Rewrite any negative exponents to reflect their position in the fraction (numerator or denominator).

This systematic approach ensures any algebra involving rational expressions becomes manageable and reduces common errors. By understanding each component deeply, dealing with complex monomials becomes a much more intuitive task.