Problem 25 - Algebra 2

Question

Simplify each expression. $$ \frac{5}{a^{2} b}-\frac{7 a}{5 a^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{25 - 7ab}{5a^{2} b} \).

Step 1: Identify the need for a common denominator

The given expression \( \frac{5}{a^{2} b} - \frac{7a}{5a^{2}} \) contains two fractions with different denominators \( a^{2}b \) and \( 5a^{2} \). To subtract these, we need a common denominator.

Step 2: Determine the least common denominator (LCD)

The denominators are \( a^{2}b \) and \( 5a^{2} \). The least common denominator should include each factor the greatest number of times it appears in any factorization. Here, the LCD is \( 5a^{2}b \).

Step 3: Rewrite each fraction with the LCD as the new denominator

Convert \( \frac{5}{a^{2} b} \) to have the denominator \( 5a^{2}b \) by multiplying the numerator and denominator by 5, giving \( \frac{25}{5a^{2} b} \). Convert \( \frac{7a}{5a^{2}} \) by multiplying the numerator and denominator by \( b \), resulting in \( \frac{7ab}{5a^{2} b} \).

Step 4: Subtract the fractions

Now that both fractions have the same denominator \( 5a^{2}b \), subtract the numerators: \( \frac{25}{5a^{2} b} - \frac{7ab}{5a^{2} b} = \frac{25 - 7ab}{5a^{2} b} \).

Step 5: Simplify if possible

The expression \( \frac{25 - 7ab}{5a^{2} b} \) cannot be simplified further as there are no common factors in the numerator and denominator that can be canceled.

Key Concepts

Least Common Denominator

When you're working with fractions, especially algebraic ones, finding a common denominator is crucial for operations like subtraction. The common denominator allows you to combine or compare fractions easily. In this exercise, we started with the fractions \( \frac{5}{a^{2} b} \) and \( \frac{7a}{5a^{2}} \), which have different denominators: \( a^{2}b \) and \( 5a^{2} \).To find the least common denominator (LCD), consider each of the denominators and determine the largest power of each factor that appears. For example:

• Both denominators include \( a^2 \), so we retain it.
• Only the first denominator includes \( b \), so we include it in the LCD.
• Only the second denominator includes 5, so we include it as well.

Thus, the LCD is \( 5a^{2}b \), which combines all unique factors.

Subtracting Fractions

Subtracting fractions requires that both fractions have the same denominator. Once you have established the least common denominator, like we did with \( 5a^{2}b \) for our fractions, you rewrite each fraction to this common denominator.For \( \frac{5}{a^{2} b} \) to have the denominator \( 5a^{2}b \), multiply both the numerator and denominator by 5. This results in \( \frac{25}{5a^{2} b} \).For \( \frac{7a}{5a^{2}} \), multiply both the numerator and denominator by \( b \). This gives you \( \frac{7ab}{5a^{2} b} \).After you convert both fractions to have the same denominator, subtract the numerators:\[\frac{25}{5a^{2} b} - \frac{7ab}{5a^{2} b} = \frac{25 - 7ab}{5a^{2} b}.\] Now you have successfully subtracted the fractions.

Simplifying Algebraic Fractions

The final step in solving any expression involving fractions is simplifying the result, which means reducing the fraction to its simplest form if possible. After subtracting the fractions, you end up with \( \frac{25 - 7ab}{5a^{2} b} \). Simplifying algebraic fractions involves checking if the numerator and denominator share any common factors. If they do, you can cancel them out.In this case, the numerator \( 25 - 7ab \) and the denominator \( 5a^{2} b \) share no common factors aside from 1. Therefore, the fraction \( \frac{25 - 7ab}{5a^{2} b} \) is already in its simplest form.When simplifying, always check for common factors carefully, but remember, not every expression can be simplified further.