Problem 57 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{\frac{b^{2}-81}{18 a^{2}}}{\frac{4 b-36}{9 a}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{b+9}{8a} \).

Step 1: Understand the Complex Fraction

A complex fraction has a numerator and a denominator that are themselves fractions. In this problem, the complex fraction is \( \frac{\frac{b^{2}-81}{18a^{2}}}{\frac{4b-36}{9a}} \). This means we need to simplify the expression by dividing the two fractions.

Step 2: Simplify Numerator and Denominator

Both the numerator and the denominator have expressions that can be factored. The numerator is \( \frac{b^{2}-81}{18a^{2}} \) and can be simplified by factoring the difference of squares: \( b^{2} - 81 = (b+9)(b-9) \). The denominator \( \frac{4b-36}{9a} \) can be factored by taking out the greatest common factor, which is 4: \( 4(b-9) \). Thus, the expression becomes \( \frac{\frac{(b+9)(b-9)}{18a^{2}}}{\frac{4(b-9)}{9a}} \).

Step 3: Set Up the Division

Dividing by a fraction is the same as multiplying by its reciprocal. So, \( \frac{\frac{(b+9)(b-9)}{18a^{2}}}{\frac{4(b-9)}{9a}} \) becomes \( \frac{(b+9)(b-9)}{18a^{2}} \times \frac{9a}{4(b-9)} \).

Step 4: Multiply the Fractions

Multiply the numerators together and the denominators together: \( \frac{(b+9)(b-9) \cdot 9a}{18a^{2} \cdot 4(b-9)} \). Now a and \( b-9 \) can be cancelled in the numerator and denominator: \( \frac{9(b+9)}{72a} \).

Step 5: Simplify Further

Simplify \( \frac{9(b+9)}{72a} \). We can factor out the greatest common factor from the numerator and denominator: \( \frac{9}{72} = \frac{1}{8} \). Thus, the expression simplifies further to \( \frac{b+9}{8a} \).

Key Concepts

Difference of Squares

The concept of a "difference of squares" is a method used to simplify expressions that are subtracting one perfect square from another. In our problem, we see this in the expression \( b^2 - 81 \). Since \( b^2 \) and \( 81 \) are both perfect squares, this expression can be rewritten using the formula:

• \( a^2 - b^2 = (a + b)(a - b) \)

For example, \( b^2 - 81 \) becomes \( (b + 9)(b - 9) \). This formula is very handy when simplifying expressions, as it breaks down a difference into a product. Recognizing and applying the difference of squares drastically simplifies the simplification process. Whenever you spot a subtraction between two perfect square terms, look to factor it using this useful formula.

Factoring

Factoring is the process of breaking down an expression into simpler terms, or "factors," that can be multiplied to result in the original expression. In the problem given, we first factor the numerator of the complex fraction by recognizing the difference of squares. Then, we factor the denominator by finding the greatest common factor (GCF).

For the denominator \( 4b - 36 \), we identify the GCF as 4. This gives us:

• \( 4b - 36 = 4(b - 9) \)

By factoring both the numerator and the denominator, the complex fraction becomes more manageable, allowing us to simplify it step-by-step. The factoring process is crucial because it transforms complicated fractions into multiplicative forms that can be easily divided or inverted when finding reciprocals.

Reciprocal of a Fraction

To understand the reciprocal of a fraction, it's important to know that it involves flipping the numerator and denominator of the fraction. This concept is key when you need to divide by a fraction. Remember that dividing by a fraction is the same as multiplying by its reciprocal.

For the fraction \( \frac{4(b-9)}{9a} \), the reciprocal is \( \frac{9a}{4(b-9)} \). Hence, when the problem requires you to divide by this fraction, you simply multiply the other fraction by \( \frac{9a}{4(b-9)} \). Flipping fractions might seem confusing at first, but with practice, it becomes a powerful tool to simplify and solve complex expressions efficiently.

Greatest Common Factor

The concept of the greatest common factor (GCF) plays a crucial role in simplifying fractions. The GCF is the biggest factor that divides two or more numbers or expressions. In our solution, we learned that the GCF of the numbers 4 and 36 is 4.

• For example, in \( 4b - 36 \), the GCF 4 is factored out, rewriting the expression as \( 4(b - 9) \).

The GCF helps in reducing fractions to their simplest form. By dividing both the numerator and denominator by their GCF, you can make expressions easier to manage, clearing the way for further simplification. This step ensures that your final answer is as simplified as possible, which is always a key goal in mathematics.