Problem 12 - Intermediate Algebra

Question

Simplify each complex fraction. See Examples 1 and 2. $$ \frac{\frac{3 b^{7}}{4}}{\frac{b^{9}}{2}} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{3}{2b^{2}}\).

Step 1: Understand the Structure of the Complex Fraction

The given expression is a complex fraction, which means it is a fraction where either the numerator, the denominator, or both contain fractions themselves. Here, \[\frac{\frac{3 b^{7}}{4}}{\frac{b^{9}}{2}}\] consists of two fractional parts: \(\frac{3 b^{7}}{4}\) in the numerator and \(\frac{b^{9}}{2}\) in the denominator.

Step 2: Rewrite as a Division of Fractions

To simplify the complex fraction, reinterpret it as dividing the numerator of the main fraction by the denominator of the main fraction. Thus, the expression becomes: \[\frac{3 b^{7}}{4} \div \frac{b^{9}}{2}.\]

Step 3: Multiply by the Reciprocal

To divide fractions, you multiply the first fraction by the reciprocal of the second fraction. The reciprocal of \(\frac{b^{9}}{2}\) is \(\frac{2}{b^{9}}\). Therefore, the expression becomes: \[\frac{3 b^{7}}{4} \times \frac{2}{b^{9}}.\]

Step 4: Multiply the Fractions

Multiply the numerators together and the denominators together:\[\frac{3b^{7} \times 2}{4 \times b^{9}}.\] Simplifying this gives:\[\frac{6b^{7}}{4b^{9}}.\]

Step 5: Simplify the Multiplication

To simplify this fraction, divide both the numerator and the denominator by their common terms. Factoring the expression:- The common factor of 6 and 4 is 2, so divide both by 2.- The common factor involving \(b\) is \(b^{7}\), so divide both by \(b^{7}\):\[\frac{6b^{7}}{4b^{9}} = \frac{3}{2b^{2}}.\]

Step 6: Final Simplified Form

The final simplified version of the original complex fraction is:\[\frac{3}{2b^{2}}.\]

Key Concepts

Fraction Simplification

When dealing with fractions, especially complex ones, the goal is to make the expression as simple as possible. Simplification helps to reduce fractions to their most basic form, making them easier to work with and understand.
The process involves:

• Finding common factors in the numerator and the denominator.
• Dividing both parts by their largest common factor.

In the context of complex fractions, which are fractions within fractions, breaking them down into simpler fractions is key. For example, when presented with something like \(\frac{6b^7}{4b^9}\), you simplify it by identifying common factors. Here, 2 is a common factor between 6 and 4, and \(b^7\) is common between the numerator and denominator. Divide these out to become \(\frac{3}{2b^2}\).
Through simplification, expressions become streamlined, laying a solid foundation for further mathematical operations.

Reciprocal

In mathematics, the concept of a reciprocal is often essential when dealing with division, especially with fractions. The reciprocal of a number is simply \(1\) divided by that number.
For a fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\). It involves switching the numerator and the denominator.

• Reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).
• Reciprocal of \(\frac{b^9}{2}\) is \(\frac{2}{b^9}\).

The reciprocal comes into play predominantly when dividing fractions. By converting the division into multiplication using the reciprocal, the process becomes straightforward. This method considerably simplifies division tasks, transforming them into multiplication, an operation deemed easier by many.

Division of Fractions

Dividing fractions might sound complicated initially, but the process is simple once you grasp the key steps. The division of fractions involves using their reciprocals to turn the division into a multiplication problem.
For instance, instead of calculating \(\frac{3b^7}{4} \div \frac{b^9}{2}\), you multiply \(\frac{3b^7}{4}\) by the reciprocal of \(\frac{b^9}{2}\), which is \(\frac{2}{b^9}\).

• Step 1: Find the reciprocal of the divisor. E.g., reciprocal of \(\frac{b^9}{2}\) is \(\frac{2}{b^9}\).
• Step 2: Change the operation from division to multiplication.
• Step 3: Multiply numerators together and multiply denominators together: \(\frac{3b^7 \times 2}{4 \times b^9}\).

By understanding and applying this method, you tackle what seems complex at first glance efficiently, minimizing errors and confusion.

Algebraic Expressions

Algebraic expressions are a mix of numbers, variables, and arithmetic operations. They form the building blocks of algebra. Understanding how to manipulate them is crucial.
In dealing with algebraic expressions, you encounter variables like \(b\) in \(3b^7\). Handling expressions with similar variables involves combining like terms and factoring when possible.

• Recognizing like terms is key. E.g., \(6b^7\) and \(4b^9\) have the common base \(b\).
• Simplifying involves canceling out terms. For instance, dividing both \(b^7\) from the numerator and \(b^9\) from the denominator results in a simpler base expression \(b^2\) in the denominator.

Through understanding algebraic expressions fully, you ease the process of integrating them into various equations and simplifying complex expressions.