Problem 22 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{7}{8 b^{2}}-\frac{5}{6 b^{3}} $$

Step-by-Step Solution

Answer

\( \frac{21b - 20}{24b^3} \)

Step 1: Identify the Problem

We need to perform the subtraction between the two fractions \( \frac{7}{8b^2} \) and \( \frac{5}{6b^3} \). This requires finding a common denominator.

Step 2: Determine the Least Common Denominator

The denominators are \( 8b^2 \) and \( 6b^3 \). Find the common denominator by taking the least common multiple of the constants and the highest power of \( b \). The least common multiple of 8 and 6 is 24, and the highest power of \( b \) is \( b^3 \). Therefore, the common denominator is \( 24b^3 \).

Step 3: Rewrite Fractions with Common Denominator

Rewrite each fraction with the denominator \( 24b^3 \).- For \( \frac{7}{8b^2} \), multiply the numerator and the denominator by \( \frac{3b}{3b} \) to get \( \frac{21b}{24b^3} \).- For \( \frac{5}{6b^3} \), multiply the numerator and the denominator by \( \frac{4}{4} \) to get \( \frac{20}{24b^3} \).

Step 4: Perform the Subtraction

Subtract the fractions: \[ \frac{21b}{24b^3} - \frac{20}{24b^3} = \frac{21b - 20}{24b^3} \].

Step 5: Simplify the Result, if Possible

The numerator \( 21b - 20 \) cannot be simplified any further with respect to the denominator \( 24b^3 \). Therefore, the result \( \frac{21b - 20}{24b^3} \) is already in its simplest form.

Key Concepts

Understanding the Least Common Denominator in Algebraic Fractions

When handling algebraic fractions, finding a common ground for the denominators of each fraction is crucial. This common ground is termed the 'least common denominator' (LCD). The least common denominator helps students perform operations like addition and subtraction on fractions by ensuring all terms share the same base.

For instance, in the fractions \( \frac{7}{8b^2} \) and \( \frac{5}{6b^3} \), the denominators are both different. To find the least common denominator:

• Identify the constants within the denominators. Here, the numbers are 8 and 6.
• Determine the least common multiple (LCM) of these numbers, which is the smallest number that both constants can evenly divide. The LCM of 8 and 6 is 24.
• Next, compare the powers of the variable \( b \). The variable \( b \) is raised to the power of 2 in one fraction and 3 in the other.
• Take the highest power for the variable, which here is \( b^3 \).

Thus, the least common denominator for our original fractions is \( 24b^3 \). Utilizing this common denominator facilitates coherent fraction manipulation.

Simplifying Algebraic Expressions

When dealing with algebraic expressions, simplification is the process of reducing them to the most straightforward form. This often involves reducing complex fractions or expressions to their simplest forms.

For the algebraic expression resulting from subtracting \( \frac{7}{8b^2} - \frac{5}{6b^3} \), you had to rewrite both fractions to share the common denominator \( 24b^3 \). This makes further operations possible:

• Multiply the numerator and denominator of \( \frac{7}{8b^2} \) by \( 3b \) to get \( \frac{21b}{24b^3} \).
• Multiply the numerator and the denominator of \( \frac{5}{6b^3} \) by 4 to get \( \frac{20}{24b^3} \).

The fraction \( \frac{21b - 20}{24b^3} \) is the simplified form of the expression post-subtraction. Here, the numerator doesn’t share any common factors with the denominator, indicating that further simplification isn't achievable. It's always essential to examine the expression post-operations to determine if further reduction is possible.

Fraction Subtraction Made Simple

Subtracting fractions, especially with algebra involved, is often challenging for many students. Essential to this process is ensuring that both fractions have the same denominator. Once the denominators match, subtraction becomes a simple arithmetic operation on the numerators.

Take the operation \( \frac{21b}{24b^3} - \frac{20}{24b^3} \), where the common denominator \( 24b^3 \) places the fractions on the same level. Subtraction is thus only concerned with:

• Directly subtracting the numerators: \( 21b - 20 \).
• Placing the result over the established common denominator: \( 24b^3 \).

This straightforward approach ensures clarity and promotes a better understanding of dealing with algebraic fractions. Approaching subtraction with a standard process encourages students align operations correctly, ensuring the expression is accurate and straightforward post-operation.