Problem 17 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{3}{5 p^{2}}-\frac{5}{10 p} $$

Step-by-Step Solution

Answer

\(\frac{6 - 5p}{10p^2}\)

Step 1: Identify the Least Common Denominator (LCD)

The two fractions have different denominators, so we need to find the least common denominator (LCD) to combine them. The denominators are \(5p^2\) and \(10p\). The least common multiple of the numerical part (5 and 10) is 10. For the variable part, the highest power of \(p\) is \(p^2\). Therefore, the LCD is \(10p^2\).

Step 2: Rewrite Each Fraction with the LCD

Convert each fraction so they have the same denominator as the LCD. For \(\frac{3}{5p^2}\), multiply both the numerator and the denominator by 2 to get the equivalent fraction: \(\frac{6}{10p^2}\). Since \(\frac{5}{10p}\) already has a denominator of 10, multiply the numerator and the denominator by \(p\) to get \(\frac{5p}{10p^2}\).

Step 3: Subtract the Fractions

Now that both fractions have the same denominator, subtract the numerators: \(\frac{6}{10p^2} - \frac{5p}{10p^2} = \frac{6 - 5p}{10p^2}\).

Step 4: Simplify, If Possible

The expression \(\frac{6 - 5p}{10p^2}\) cannot be simplified further as there are no common factors in the numerator and the denominator. Hence, this is the simplest form.

Key Concepts

Least Common Denominator

When simplifying algebraic fractions, finding the least common denominator (LCD) is crucial. The LCD is essentially a common denominator that allows you to combine fractions easily. It should be the smallest possible value that can evenly divide all the denominators in question.
In algebraic expressions, it's instrumental because the denominators can involve variables, not just numbers. To find the LCD in such cases, you must consider both the numerical and the variable parts:

• Numerical Part: Identify the least common multiple (LCM) of the numbers in the denominators. For example, if the denominators have numbers 5 and 10, the LCM is 10.
• Variable Part: Use the highest power of the variable present in the denominators. If you have denominators like \(5p^2\) and \(10p\), the variable part of the LCD will be \(p^2\).

By combining these two results, you get the complete LCD, which in this exercise is \(10p^2\). This ensures that all terms are aligned for the arithmetic operations.

Combining Fractions

Once you have a common denominator, combining fractions becomes straightforward. Here's how it works step by step:
Convert each fraction to its equivalent with the LCD. This requires multiplying the numerator and denominator by a specific factor that makes the denominator equal the LCD. Let's unravel how you achieve this:

• First Fraction: For \(\frac{3}{5p^2}\), we need to transform it into a fraction whose denominator matches \(10p^2\). Multiply both the numerator and denominator by 2 to achieve \(\frac{6}{10p^2}\).
• Second Fraction: For \(\frac{5}{10p}\), multiply both the numerator and the denominator by \(p\) to get \(\frac{5p}{10p^2}\).

With both fractions now expressing their numerators over a common denominator, you can directly subtract the numerators: \(\frac{6}{10p^2} - \frac{5p}{10p^2}\). Keep the common denominator while performing the operation.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operations. When dealing with algebraic fractions, several elements interact:

• Numerators and Denominators: Each fraction has a numerator and a denominator, which can involve constants and variables.
• Operations: Often, you need to add, subtract, multiply, or divide these fractions. The key is ensuring a common denominator for addition and subtraction.
• Variables: In our case, the variable \(p\) appears in the denominators. We must consider its role when determining the least common denominator.

After performing operations such as combining the numerators in this exercise, it's wise to simplify the expression. In this instance, \(\frac{6 - 5p}{10p^2}\) is already in its simplest form since there are no common factors between the numerator and the denominator. Recognizing when further simplification is not possible is an essential skill in handling algebraic expressions effectively.