Problem 52 - Intermediate Algebra

Question

Divide, and then simplify, if possible. See Example 6. $$ \frac{24 a^{6}}{b} \div \frac{64 a^{9}}{b^{2}} $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{3 b}{8 a^{3}} \).

Step 1: Rewrite the Division Problem

Convert the division of fractions into multiplication by the reciprocal. The problem \( \frac{24 a^{6}}{b} \div \frac{64 a^{9}}{b^{2}} \) becomes \( \frac{24 a^{6}}{b} \times \frac{b^{2}}{64 a^{9}} \).

Step 2: Simplify by Cancelling Common Factors

Look for common factors both in numerators and denominators. Cancel out the common factors: \( b \) cancels with one \( b \) from \( b^2 \), and factors of 2 in 24 and 64. \( \frac{24}{64} = \frac{3}{8} \). The terms in \( a \) can be simplified: \( a^{6} / a^{9} = 1 / a^{3} \).

Step 3: Combine the Simplified Fractions

Combine the fractions. After simplifying, we are left with \( \frac{3 b}{8 a^{3}} \).

Step 4: Verify the Result

Check the result by ensuring division of variables and constants is simplified, and no further simplifications are possible.

Key Concepts

Simplification

Simplification is an important process in algebra that makes expressions easier to work with and understand. In algebraic expressions, simplification often involves reducing fractions, cancelling out common factors, and rewriting expressions in a more manageable form.

When simplifying, we aim to find the simplest equivalent form of the expression. This often involves the following steps:

• Identify and cancel any like terms or common factors between the numerators and denominators.
• Reduce numbers to their simplest form, much like simplifying ordinary numerical fractions.
• Rewrite variables raised to powers by using the laws of exponents to combine or cancel them.

By breaking down a complex fraction into simpler parts, we make it more intuitive to both evaluate and understand. This process allows us to work more efficiently with algebraic expressions.

Fractions

Fractions represent a central theme in mathematics where a whole is divided into equal parts. In general, they are written in the form \( \frac{a}{b} \), where "a" is the numerator indicative of how many parts we have, and "b" is the denominator indicating into how many parts the whole is divided.

In algebra, fractions can contain not only numbers but also variables and their powers. Understanding fractions involves:

• Knowing how to perform basic operations like addition, subtraction, multiplication, and division.
• Recognizing the need to find a common denominator for addition and subtraction.
• Understanding how to multiply and divide by flipping the fraction and multiplying, known as finding the reciprocal.

For the given exercise, converting division into multiplication of a reciprocal gives us a more straightforward path for simplification and resolution.

Algebraic Fractions

Algebraic fractions are fractions in which the numerator, denominator, or both contain algebraic expressions. They function similarly to regular fractions but require additional steps for manipulation because they involve variables.

Key points to consider when working with algebraic fractions include:

• Converting division of fractions into multiplication by inversing (reciprocal) the second fraction.
• Utilizing exponent laws like \( a^m / a^n = a^{m-n} \) to simplify variables within fractions.
• Ensuring that we perform the same operation across both numerators and denominators during simplification, maintaining equivalence.

In the example exercise, converting the division to multiplication made it easier to cancel common terms, leading to a clear simplified form. Thus, understanding these principles helps students effectively decode and manipulate algebraic fractions.