Problem 51 - College Algebra with Modeling and Visualization

Question

Exercises \(45-54:\) Use the quotient rule to simplify the expression. Use positive exponents to write your answer. $$ \frac{12 a^{2} b^{3}}{18 a^{4} b^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{2b}{3a^2}\).

Step 1: Identify the Quotient Rule

The quotient rule for simplifying fractions states that the exponent in the denominator is subtracted from the exponent in the numerator for like bases. That is, if you have a fraction \(\frac{x^m}{x^n}\), it simplifies to \(x^{m-n}\).

Step 2: Simplify Coefficients

Divide the coefficients 12 and 18. The greatest common divisor of 12 and 18 is 6, so \(\frac{12}{18} = \frac{2}{3}\).

Step 3: Simplify the 'a' component

Apply the quotient rule to the 'a' component: \(a^2\) in the numerator and \(a^4\) in the denominator gives \(a^{2-4}=a^{-2}\).

Step 4: Simplify the 'b' component

Apply the quotient rule to the 'b' component: \(b^3\) in the numerator and \(b^2\) in the denominator gives \(b^{3-2}=b^{1}\).

Step 5: Write the expression with positive exponents

Substitute the simplified parts back into the expression: \(\frac{2}{3} \cdot a^{-2} \cdot b^{1}\). Convert \(a^{-2}\) to positive exponents: \(\frac{2b}{3a^2}\).

Step 6: Combine the components

The simplified expression is: \(\frac{2b}{3a^2}\).

Key Concepts

Exponent Rules

Exponent rules are essential when working with algebraic expressions, especially in operations involving multiplication and division of terms with the same base.

• Basic Rule: When multiplying similar bases, add the exponents. For example, \(x^m \times x^n = x^{m+n}\).
• Quotient Rule: When dividing similar bases, subtract the exponents. If you have \(\frac{x^m}{x^n}\), it simplifies to \(x^{m-n}\). This rule helps to simplify expressions by reducing the powers of common bases.
• Negative Exponent Rule: Any base with a negative exponent moves to the other part of the fraction as a positive exponent. For instance, \(x^{-n} = \frac{1}{x^n}\).

Using these rules can make solving problems with complex fraction expressions more straightforward. They ensure the expression is simplified as much as possible, often resulting in easier calculations and an understandable final form.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This process is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD). Here's a quick rundown:

• Finding the GCD: Identify the largest number that divides both numerator and denominator without leaving a remainder. For instance, with the fraction \(\frac{12}{18}\), the GCD is 6.
• Divide Both Parts: Divide both the numerator and denominator by the GCD. Continuing with \(\frac{12}{18}\), when divided by 6, it reduces to \(\frac{2}{3}\).
• Final Check: Ensure no further common divisors exist beyond 1, indicating the fraction is fully simplified.

This process is crucial as it transforms the fraction into a simpler form, making calculations easier and faster. Simplified fractions are especially important in algebra, leading to cleaner expressions and more manageable equations.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operation symbols. Understanding how to work with these components forms the foundation for simplifying expressions and solving equations.
Key elements of algebraic expressions include:

• Variables: Symbols like \(x\), \(a\), or \(b\) that represent unknown values. They can change and hold different values in different contexts.
• Constants: Fixed numbers like 12, 18, or any integer. They do not change within the scope of the problem.
• Operations: Include addition, subtraction, multiplication, and division, dictating how the terms in the expression interact.
• Terms: Combinations of variables and constants like \(12a^2b^3\), which are parts of the entire expression separated by addition or subtraction. Each term consists of factors multiplied together.

Understanding these basics aids in manipulating and simplifying expressions, as well as applying rules like the quotient rule to achieve the simplest form. The right handling of variables and constants transforms complex expressions into manageable and solvable algebraic problems.