Problem 35 - Elementary and Intermediate Algebra

Question

Simplify each rational expression. See Example 3 . $$\frac{24 x^{3} y^{4}}{54 x^{4} y^{3}}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{4y}{9x} \).

Step 1: Identify the Greatest Common Factor (GCF) of Coefficients

Identify the greatest common factor between the coefficients of the numerator and the denominator. The coefficients are 24 (numerator) and 54 (denominator). The GCF of 24 and 54 is 6.

Step 2: Simplify the Coefficient by Dividing by GCF

Divide both the numerator and the denominator by their GCF: \( \frac{24}{6} = 4 \) and \( \frac{54}{6} = 9 \). The expression becomes \( \frac{4 x^{3} y^{4}}{9 x^{4} y^{3}} \).

Step 3: Simplify the Powers of x

Subtract the exponent in the denominator from the exponent in the numerator for the \( x \) terms: \( x^{3-4} = x^{-1} \). This simplifies to \( \frac{1}{x} \). The expression becomes \( \frac{4 y^{4}}{9 x y^{3}} \).

Step 4: Simplify the Powers of y

Subtract the exponent in the denominator from the exponent in the numerator for the \( y \) terms: \( y^{4-3} = y^{1} \). This simplifies to \( y \). The expression becomes \( \frac{4 y}{9 x} \).

Step 5: Write the Final Simplified Expression

The simplified form of the given rational expression is \( \frac{4 y}{9 x} \).

Key Concepts

Greatest Common Factor

Understanding the Greatest Common Factor (GCF) is essential for simplifying rational expressions. When you look at fractions with integers, the GCF is the largest integer that divides both numbers without leaving a remainder. For example, in the expression \( \frac{24x^3y^4}{54x^4y^3} \), we focus first on the coefficients 24 and 54.
To find the GCF:

• List the factors of each coefficient.
• For 24, the factors are: 1, 2, 3, 4, 6, 8, 12, 24.
• For 54, the factors are: 1, 2, 3, 6, 9, 18, 27, 54.
• The common factors are 1, 2, 3, and 6. The greatest of these is 6.

Once the GCF is determined, divide both the numerator and denominator by this value. This step simplifies the expression, making further simplification easier.

Exponent Rules

Understanding exponent rules is crucial when simplifying rational expressions involving variables with exponents. Rules for exponents provide a standardized way to handle powers. Here are some key rules:

• Product Rule: When multiplying two powers with the same base, add the exponents: \( a^m \cdot a^n = a^{m+n} \).
• Quotient Rule: When dividing, subtract the exponents: \( a^m / a^n = a^{m-n} \).
• Power Rule: When raising a power to a power, multiply the exponents: \( (a^m)^n = a^{m \times n} \).

For the expression \( \frac{4x^3y^4}{9x^4y^3} \), apply the quotient rule to subtract exponents when reducing similar variables. Subtract the exponent of the denominator from the numerator for both \( x \) and \( y \). This ensures each variable is in its simplest form.

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Simplifying rational expressions follows the same principles as simplifying numeric fractions, but with the added complexity of variables.
To efficiently simplify a rational expression:

• Start by identifying and factoring the polynomials, if possible.
• Determine the GCF of numbers and variables as previously mentioned.
• Apply the exponent rules to reduce like terms.
• Always check your final expression to verify all terms are simplified thoroughly.

For example, in simplifying \( \frac{24x^3y^4}{54x^4y^3} \), we combine finding the GCF and using exponent rules for an effective simplification. Reducing each element helps achieve the final simplest form with fewest terms.

Fraction Simplification

Simplifying fractions is all about making the numerator and the denominator as simple as possible while retaining the value of the expression. When a fraction consists of both numbers and variables, simplify each component:

• Numerical Coefficients: Use the GCF to reduce the numerical coefficients by dividing, just like numeric fractions.
• Variable Terms: Use exponent rules to combine like terms and simplify.

Take the expression \( \frac{4y}{9x} \). It has been simplified step by step:

• The coefficients were first reduced using their GCF (6 in this case).
• The exponents were simplified using subtraction, applying the exponent rules.
• The final expression retains minimal numeric and variable terms, showing simplicity and accuracy.

Understanding and applying these simplification methods makes solving complex rational expressions more manageable and less error-prone.