Problem 8 - Intermediate Algebra

Question

Complete each solution to simplify the rational expression. $$ \frac{\frac{2}{a}-\frac{1}{b}}{\frac{5}{a}+\frac{3}{b}}=\frac{\frac{2}{a}-\frac{1}{b}}{\frac{5}{a}+\frac{3}{b}} \cdot \frac{\square}{\square} $$ $$ =\frac{\left(\frac{2}{a}-\frac{1}{b}\right)}{\left(\frac{5}{a}+\frac{3}{b}\right)} \frac{\square}{\square} $$ $$ =\frac{\frac{2}{a} \cdot \square}{\frac{5}{a} \cdot \square}+\frac{\frac{1}{b} \cdot \square}{b \cdot \square} $$ $$ =\frac{2 b- \square}{\square +3 a} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{2b - a}{5b + 3a}\).

Step 1: Identify Common Denominators

To simplify the expression, we need to eliminate the fractions within the larger fraction. Observe that the denominators within the numerator and denominator are \(a\) and \(b\).

Step 2: Multiply by Reciprocal for Simplification

Multiply both the numerator and denominator of the large fraction by \(ab\), the product of the denominators within the smaller fractions. This will eliminate the fractions inside the big fraction.

Step 3: Apply Multiplication to Numerator and Denominator

Multiply the numerator \(\left(\frac{2}{a} - \frac{1}{b}\right)\) by \(ab\), resulting in \((2b - a)\). Similarly, multiply the denominator \(\left(\frac{5}{a} + \frac{3}{b}\right)\) by \(ab\), resulting in \((5b + 3a)\).

Step 4: Substitute Back in the Fraction

Substitute back the simplified numerator and denominator into the fraction to get: \[\frac{2b - a}{5b + 3a}\].

Key Concepts

Algebraic Fractions

Algebraic fractions are similar to regular fractions, but they contain variables in addition to numbers. These variables make algebraic fractions capable of representing a broader range of expressions.
For instance, in the expression \( \frac{2}{a} \), \(a\) is a variable and the denominator of the fraction. These algebraic fractions follow many of the same arithmetic rules as regular fractions, such as having a numerator (the top part) and a denominator (the bottom part).
When working with algebraic fractions, it is crucial to remember that you can manipulate them using techniques like combining or simplifying, which often involves factoring or finding common denominators. This allows you to rewrite expressions in simpler forms.

Common Denominators

To simplify algebraic fractions within a larger expression, identifying common denominators is a key step. By finding a common denominator, you can combine or compare fractions more easily.
In the original exercise, the fractions \( \frac{2}{a} \) and \( \frac{1}{b} \) have different denominators, \(a\) and \(b\). To simplify, you can multiply each fraction by a form of 1 that includes the opposite fraction's denominator, creating a common base: \(ab\).
This is because any number divided by itself equals 1, e.g., \( \frac{b}{b} = 1 \). By ensuring both fractions have \(ab\) as their denominator, you set the stage for further simplification into a more manageable form.

Multiplication by Reciprocal

Multiplying by a reciprocal is a powerful algebraic tool, especially useful when simplifying complex rational expressions. A reciprocal of a number or fraction is simply its inverse, for example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). Multiplying by a reciprocal effectively cancels out the original number.
In the stated solution, multiplying both the numerator and denominator of the larger fraction by \(ab\), the reciprocal of the smaller fractions' denominators, simplifies the expression. This operation removes the inner fractions by converting them into whole numbers, making the entire expression easier to work with.
This step is critical, as it transforms the problem into one that is more straightforward to evaluate and simplifies the entire algebraic expression without altering its value.

Rational Expressions Simplification

The simplification of rational expressions involves reducing them to their simplest form. In the context of the original problem, after dealing with common denominators and multiplying by the reciprocal, you arrived at simpler expressions like \(2b - a\) for the numerator and \(5b + 3a\) for the denominator.
Simplifying means eliminating complex fractions, canceling like terms, or factoring, which leads to an expression that is easier to interpret or solve within equations.
Once reduced to the simplest terms, the rational expression \(\frac{2b - a}{5b + 3a}\) is the final, simplest form of the given algebraic fraction, and can be used to find solutions to related algebra problems. Remember, keeping expressions as simplified as possible helps in understanding the problem better and often simplifies subsequent calculations or applications.