Problem 43 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{6 b}{b+2} \div \frac{3 b^{4}}{2 b+4} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{4}{b^3} \).

Step 1: Understand Division of Fractions

When dividing fractions, the rule is to multiply by the reciprocal. In this case, we have a division between two fractions \( \frac{6b}{b+2} \) and \( \frac{3b^4}{2b+4} \). To proceed, we must convert this division into a multiplication by flipping the second fraction.

Step 2: Reciprocal of the Second Fraction

Find the reciprocal of the second fraction \( \frac{3b^4}{2b+4} \), which is \( \frac{2b+4}{3b^4} \). Now the expression becomes a multiplication: \( \frac{6b}{b+2} \times \frac{2b+4}{3b^4} \).

Step 3: Simplify the Expression by Canceling Common Factors

First, notice that \( 2b+4 \) can be factored as \( 2(b+2) \). Substituting this into the expression gives us \( \frac{6b}{b+2} \times \frac{2(b+2)}{3b^4} \). Cancel out the common factor \( b+2 \) from the numerator and the denominator, simplifying the expression to \( \frac{6b \cdot 2}{3b^4} \).

Step 4: Simplify by Performing Multiplication

Perform the multiplication across the numerators and denominators: \( \frac{12b}{3b^4} \).

Step 5: Final Simplification by Dividing Common Factors

Divide both the numerator and denominator by 3 to get \( \frac{4b}{b^4} \). Simplify further by canceling \( b \) in the numerator with one of the \( b \) terms in the denominator: \( \frac{4}{b^3} \).

Key Concepts

Simplifying Expressions

Simplifying expressions is an important concept in algebra that helps to make complex expressions more manageable. When you simplify an expression, you aim to reduce it to its simplest form while retaining its original value.

If you have a complex expression, like the algebraic fraction in this exercise, you must follow a series of steps to simplify it correctly.

• Identify factors that appear both in the numerator and the denominator.
• Use mathematical operations judiciously to eliminate these factors.

To illustrate, in algebraic expressions involving fractions, identifying common factors is key to simplification. Simplifying can involve distributing, factoring, or using basic operations like addition and subtraction to consolidate terms.

Dividing Fractions

Dividing fractions can often seem tricky, but with a clear strategy, it becomes straightforward. A fundamental rule when dividing any fraction is to multiply by the reciprocal.

• The reciprocal of a fraction is simply achieved by swapping its numerator and denominator.
• Reversing the second fraction in a division turns the operation into multiplication.

For example, in our exercise, we start with the division \[\frac{6b}{b+2} \div \frac{3b^4}{2b+4}\]and convert it via reciprocal to a multiplication:\[\frac{6b}{b+2} \times \frac{2b+4}{3b^4}\]This approach simplifies handling fractions substantially, as multiplication is generally easier to perform than division.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into simpler components or 'factors.' This step is crucial when dealing with algebraic expressions as it can reveal common factors that you might cancel out later.

• Look for the greatest common factor (GCF) in polynomial terms.
• Break down composite terms into products of simpler terms.

In the current exercise, we identified that the polynomial \(2b+4\) can be factored into \(2(b+2)\). This step is essential to simplify the overall expression further. Recognizing such patterns can greatly enhance your problem-solving efficiency when working with polynomials.

Canceling Common Factors

Canceling common factors is a method that simplifies fractions by eliminating terms that appear both in the numerator and the denominator. This step reduces clutter and focuses on the essential parts of a fraction.

When you cancel out these common factors, you essentially reduce the fraction to its simplest terms, which are often easier to interpret and work with.

• Ensure that the term being canceled is indeed a factor of the entire numerator and denominator.
• An overlooked step might result in an incorrect simplification.

In our expression, after factoring, we managed to cancel \(b+2\), which appears in both the numerator and denominator. Similarly, recognizing \(b\) as a common factor allowed for further simplification from \(\frac{4b}{b^3}\) to \(\frac{4}{b^3}\). Careful cancellation is a powerful strategy to simplify, emphasizing the final expression's underlying simplicity.