Problem 45 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{3 a+1}{a^{7}} \div \frac{a+1}{3 a^{8}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3a^2 + 3a}{a^7 + a^8} \).

Step 1: Rewrite the Division as Multiplication

To simplify the expression \( \frac{3a + 1}{a^7} \div \frac{a + 1}{3a^8} \), we first rewrite the division as multiplication by the reciprocal. This changes the expression to \( \frac{3a + 1}{a^7} \times \frac{3a^8}{a + 1} \).

Step 2: Simplify by Cancelling Common Factors

Next, we examine the factors in the numerator and denominator. Notice that \((a+1)\) and \(3a + 1\) are unique and cannot be canceled directly, but the \(a^7\) in the denominator of the first fraction can be partially canceled with \(a^8\) in the numerator of the reciprocal, leaving \(a\) in the numerator. Thus, the expression becomes \( \frac{(3a + 1) \cdot 3a}{(a^7) \cdot (a + 1)} \).

Step 3: Rearrange and Finalize the Simplification

Combine the numerators and denominators, factoring out and simplifying wherever possible. The expression becomes \( \frac{3a (3a + 1)}{a^7 (a + 1)} \). Since no further terms can be canceled, the expression is now fully simplified to \( \frac{3a^2 + 3a}{a^7 + a^8} \), recognizing that \(a\) and \((a + 1)\) remain in separate factors.

Key Concepts

Simplification

Simplification of algebraic expressions, especially involving fractions, is about reducing them to their simplest forms. This involves:

• Identifying common terms in both the numerator and the denominator that can be simplified or canceled.
• Looking for greatest common factors and terms that repeat.

In the exercise, we have an expression that requires simplification, beginning with rewriting division as multiplication, which can sometimes help clarify which terms may cancel more easily. By examining each term carefully, you ensure nothing important is left out while simplifying the expression. Typically, a simplified expression is much easier to solve or integrate into more complex equations, making this step particularly crucial in algebra.

Division of Fractions

When dividing fractions, the key is to understand that division can be transformed into multiplication. This is achieved by

• Turning the division sign into a multiplication sign.
• Flipping the second fraction (also known as taking its reciprocal).

For instance, dividing by \( \frac{a+1}{3a^8} \) involves flipping it to \( \frac{3a^8}{a+1} \) and then multiplying. This method isn't just limited to numbers - it works perfectly well with variables and algebraic terms too! By converting the division into a multiplication problem, you often make it easier to cancel and simplify terms, which paves the path for the simpler expression that follows.

Multiplication of Fractions

Multiplying fractions is straightforward once you've got the hang of the basic steps:

• Multiply the numerators together to get the new numerator.
• Multiply the denominators together to get the new denominator.

In this operation, it's essential to keep track of all terms. For algebraic fractions, this means carefully working with each variable and coefficient. The trickier aspect often involves managing the variables, especially when they come with exponents, like \(a^7\) and \(a^8\). After multiplying, look for spots where you can factor and simplify further—often this means canceling out common terms in the numerator and denominator. This step reduces complex fractions to their simplest terms, which can significantly simplify future computations.