Problem 56 - College Algebra with Corequisite Support

Question

For the following exercises, perform the given operations and simplify. \(\frac{\frac{4 a+1}{2 a-3}+\frac{2 a-3}{2 a+3}}{\frac{4 a^{2}+9}{a}}\)

Step-by-Step Solution

Answer

The simplified result is \( \frac{2a(6a^2 + a + 6)}{(2a-3)(2a+3)(4a+9)} \).

Step 1: Analyze the Expression Components

The given expression is a complex fraction: \( \frac{\frac{4a+1}{2a-3} + \frac{2a-3}{2a+3}}{\frac{4a^2+9}{a}} \). It consists of three main parts: the numerator which itself is a sum of two fractions, and the denominator which is a single fraction.

Step 2: Find a Common Denominator for the Numerator

The numerator is \( \frac{4a+1}{2a-3} + \frac{2a-3}{2a+3} \). To add these fractions, find a common denominator. The least common denominator (LCD) is \((2a-3)(2a+3)\). Rewrite each fraction with this common denominator: \( \frac{(4a+1)(2a+3) + (2a-3)(2a-3)}{(2a-3)(2a+3)} \).

Step 3: Expand the Numerator

Expand the expressions in the numerators:- \((4a+1)(2a+3) = 8a^2 + 12a + 2a + 3 = 8a^2 + 14a + 3\)- \((2a-3)(2a-3) = 4a^2 - 6a - 6a + 9 = 4a^2 - 12a + 9\)Adding these results together:\(8a^2 + 14a + 3 + 4a^2 - 12a + 9 = 12a^2 + 2a + 12\)

Step 4: Simplify the Numerator

After combining like terms, the numerator simplifies to \( \frac{12a^2 + 2a + 12}{(2a-3)(2a+3)} \). Factor out 2 from the numerator: \( \frac{2(6a^2 + a + 6)}{(2a-3)(2a+3)} \).

Step 5: Simplify the Denominator

The denominator is \( \frac{4a^2 + 9}{a} \). Rewrite it as \( \frac{4a^2}{a} + \frac{9}{a} = 4a + \frac{9}{a} \). This is a simplification of each term individually by dividing by 'a'.

Step 6: Combine the Complex Fraction

Rewrite the entire expression by substituting the simplified forms into the complex fraction:\( \frac{\frac{2(6a^2 + a + 6)}{(2a-3)(2a+3)}}{4a + \frac{9}{a}} \). Rewrite this as ordinary division of fractions:\( \frac{2(6a^2 + a + 6)}{(2a-3)(2a+3)} \times \frac{a}{(4a^2 + 9)} \).

Step 7: Simplify the Expression

Perform the multiplication of the fractions:\[ \frac{2a(6a^2 + a + 6)}{(2a-3)(2a+3)(4a+9)} \].Verify any further simplification by checking if \(6a^2 + a + 6\) has common factors, which it doesn't clearly show.

Step 8: Final Simplified Form

The expression is fully simplified and the final form is:\( \frac{2a(6a^2 + a + 6)}{(2a-3)(2a+3)(4a+9)} \). There are no further reductions possible without additional factoring potential, concluding simplification.

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing expressions into their simplest form. To simplify an expression, you need to understand various components: variables, coefficients, terms, and operations involved. The process involves adding and subtracting like terms, which are terms with the same variable raised to the same power. This makes it easier to perform operations like addition and subtraction.

For example, in the expression \(8a^2 + 14a + 3 + 4a^2 - 12a + 9\), you can group and combine like terms: - Combine terms with \(a^2\) which results in \(12a^2\).- Combine terms with \(a\) resulting in \(2a\).- Combine constant terms to \(+ 12\).

The simplified form becomes \(12a^2 + 2a + 12\). Using simplification methods, expressions become easier to interpret and solve.

Common Denominator

When dealing with complex fractions, finding a common denominator is crucial for addition or subtraction. The common denominator is a shared multiple of the denominators of the fractions involved. It allows you to combine fractions by aligning them under the same benchmark.

In the numerator of our fraction, \(\frac{4a+1}{2a-3} + \frac{2a-3}{2a+3}\), we determine the least common denominator (LCD) to be \((2a-3)(2a+3)\). This is found by multiplying the denominators of each fraction together.

• This allows each fraction to be rewritten as \(\frac{(4a+1)(2a+3) + (2a-3)(2a-3)}{(2a-3)(2a+3)}\), aligning them under the common denominator for addition.

Finding a common denominator is essential as it allows us to effectively add or subtract fractions.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler, multiplied polynomials that represent the same value. This is a key step in algebra because it simplifies expressions and solves equations.

In our problem, after expanding and combining like terms in the numerator \(12a^2 + 2a + 12\), the expression is simpler to handle. A factor that could be extracted is 2, leading to:

• The expression becomes \(2(6a^2 + a + 6)\).

Factoring becomes more crucial when further simplification could reduce the terms further, although in this instance, \(6a^2 + a + 6\) was not factorable further without any apparent common factors.

This concept is key in simplifying complex fractions and solving equations.

Fraction Multiplication

Multiplying fractions is an essential arithmetic skill, particularly when dealing with complex fractions. It involves multiplying the numerators together and the denominators together. Simplifying or factoring fractions before multiplying can make this process easier.

For the complex fraction \(\frac{\frac{2(6a^2 + a + 6)}{(2a-3)(2a+3)}}{4a + \frac{9}{a}}\), convert it to multiplication by the inverse of the denominator fraction, leading to:

• Multiply as \(\frac{2(6a^2 + a + 6)}{(2a-3)(2a+3)} \times \frac{a}{4a^2 + 9}\).

- You only need to multiply straight across: - Numerators: \(2a(6a^2 + a + 6)\) - Denominators: \((2a-3)(2a+3)(4a + 9)\).

This straightforward method of multiplication is important for systematically working through and simplifying complex algebraic expressions.