Problem 17 - Elementary Algebra

Question

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{a^{2}+6 a+5}{6 a}}{\frac{a^{2}-1}{24 a^{4}}} $$

Step-by-Step Solution

Answer

\(\frac{4a^{3}(a+5)}{a-1}\)

Step 1 - Simplify the Numerator and Denominator Separately

First, simplify the numerator \(\frac{a^{2}+6a+5}{6a}\) and the denominator \(\frac{a^{2}-1}{24a^{4}}\)

Step 2 - Factor the Numerator and Denominator

Factor both expressions completely:\[a^{2}+6a+5 = (a+1)(a+5)\]and\[a^{2}-1 =(a+1)(a-1)\]

Step 3 - Substitute the Factors

Substitute the factored forms into the expression:\[\frac{\frac{(a+1)(a+5)}{6a}}{\frac{(a+1)(a-1)}{24a^{4}}}\]

Step 4 - Simplify the Complex Fraction

Rewrite the complex fraction as a division problem:\[\frac{(a+1)(a+5)}{6a} \div \frac{(a+1)(a-1)}{24a^{4}}\]This equals:\[\frac{(a+1)(a+5)}{6a} \times \frac{24a^{4}}{(a+1)(a-1)}\]

Step 5 - Cancel Common Factors

Cancel the common factors in the numerator and denominator:\(a+1\) cancels out, and so do the factors of \(a\):\[\frac{(a+5) \cdot 24a^{3}}{6(a-1)}\]

Step 6 - Simplify the Result

Simplify the remaining terms:\[\frac{24a^{3}(a+5)}{6(a-1)}\]Divide the constants and get:\[4a^{3}(a+5)/(a-1)\]

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra. It involves breaking down a polynomial into simpler 'factors' that, when multiplied together, give you the original polynomial. Consider the expression \(a^2 + 6a + 5\). To factor it, we need to find two numbers that multiply to 5 (the constant term) and add up to 6 (the coefficient of the middle term).
The factors are \((a+1)(a+5)\). Similarly, for the expression \(a^2 - 1\), we use the difference of squares method. In this case, we can write it as \((a+1)(a-1)\).
Factoring helps to simplify complex mathematical expressions and is a powerful tool in solving equations and simplifying rational expressions.

Complex Fractions

A complex fraction contains a fraction in its numerator, denominator, or both. In our exercise, we have:

• Numerator: \(\frac{a^2+6a+5}{6a}\)
• Denominator: \(\frac{a^2-1}{24a^4}\)

To simplify complex fractions, follow these steps:
1. Simplify the numerator and denominator individually.
This means factoring them first, as we did with \((a+1)(a+5)\) and \((a+1)(a-1)\).
2. Rewrite the complex fraction as a division problem. In our case, it becomes \(\frac{(a+1)(a+5)}{6a}\) ÷ \(\frac{(a+1)(a-1)}{24a^4}\).
The division of fractions is handled by multiplying the first fraction by the reciprocal of the second fraction.

Canceling Common Factors

After rewriting the complex fraction as a division problem, we multiply and then cancel common factors to simplify.For example:

• First, rewrite the expression: \(\frac{(a+1)(a+5)}{6a} \times \frac{24a^4}{(a+1)(a-1)}\)
• Next, cancel out common factors in the numerator and denominator.

Here, \((a+1)\) appears in both the numerator and denominator, and so does \(a\). Canceling these factors simplifies the expression. The result is \(\frac{(a+5) \times 24a^3}{6(a-1)}\). This step is crucial because it makes the expression much simpler and easier to handle.

Multiplication of Fractions

In our final steps, we're dealing with the multiplication of fractions. The goal is to simplify to the lowest terms.After canceling the common factors, we have:\(\frac{(a+5) \times 24a^3}{6(a-1)}\)
To simplify further, follow these steps:
1. Multiply the remaining factors in the numerator: \((a+5) \times 24a^3\).
2. Multiply any constants and variables.
Our result is \(24a^3(a+5)\).
Last, divide 24 by 6 to get 4, which simplifies to \(4a^3(a+5)\). The simplified expression is \(\frac{4a^3(a+5)}{a-1}\).
The key to multiplying fractions is first canceling common factors, then simplifying by multiplying the remaining terms.