Problem 82 - Elementary Algebra

Question

For the following problems, convert the given rational expressions to rational expressions having the same denominators. $$ \frac{10 a}{a-6}, \frac{2}{a^{2}-6 a} $$

Step-by-Step Solution

Answer

Question: Rewrite the given rational expressions with the same denominator: \(\frac{10a}{a-6}\) and \(\frac{2}{a^2-6a}\). Answer: The equivalent rational expressions with the same denominator are \(\frac{10a^2}{a(a-6)}\) and \(\frac{2}{a(a-6)}\).

Step 1: Identify the denominators of both expressions

The denominators of the given rational expressions are \((a-6)\) and \((a^2 - 6a)\).

Step 2: Find the lowest common denominator (LCD)

To find the LCD, notice that \((a^2 - 6a)\) can be factored as \(a(a - 6)\). Therefore, the LCD is \(a(a - 6)\).

Step 3: Rewrite the first expression with the LCD

In order to rewrite the first expression, we need to multiply the numerator and the denominator by the missing factor. Since the denominator of the first expression is \((a-6)\), we need to multiply both the numerator and the denominator by \(a\). So, we have: $$ \frac{10a}{a - 6} = \frac{10a \cdot a}{(a - 6) \cdot a} = \frac{10a^2}{a(a-6)} $$

Step 4: Rewrite the second expression with the LCD

The denominator of the second expression is already the same as the LCD, so we don't need to do anything to it. It remains: $$ \frac{2}{a^2-6a} = \frac{2}{a(a-6)} $$ The rational expressions with the same denominators are: $$ \frac{10a^2}{a(a-6)} \text{ and } \frac{2}{a(a-6)} $$

Key Concepts

common denominator

When working with rational expressions, it's essential to find a common denominator to add, subtract, or compare them effectively.
Rational expressions, like fractions, must share the same denominator to perform operations on them without changing their value.
Here's why this is important and how you can find a common denominator:

• **Understanding Denominators**: The denominator in a rational expression indicates the total number of equal parts a whole is divided into. For expressions \(\frac{10a}{a-6} \) and \(\frac{2}{a^2-6a} \), they have different denominators, \((a-6)\) and \((a^2 - 6a)\).
• **Finding the Least Common Denominator (LCD)**: To find a common denominator, determine the least common multiple of both denominators. Here, \((a^2 - 6a)\) can be factored as \(a(a-6)\), making \(a(a-6)\) the LCD.
• **Rewriting Expressions**: After finding the LCD, adjust each expression by multiplying its numerator and denominator to convert to the common denominator. This ensures that both expressions share the same terms in the denominator.

Finding a common denominator is pivotal in simplifying expressions, solving problems, and maintaining equality.

factoring polynomials

Factoring polynomials is a crucial skill because it simplifies expressions and solves equations more efficiently.
In the context of rational expressions, factoring helps us find common denominators and simplify terms.

• **What is Factoring?**: Factoring involves breaking down a polynomial into simpler "factors" that, when multiplied together, give back the original polynomial.
  For instance, the polynomial \(a^2 - 6a\) can be factored to \(a(a - 6)\).
• **Identifying Factors**: Look for common terms across a polynomial's terms, allowing you to segregate them into their factors.
  In our example, both terms \((a^2\) and \(- 6a)\) share an \(a\), which is pulled out to simplify the expression.
• **Factoring Simplifies Solutions**: This step is key to finding the least common denominator, needed to solve problems involving multiple rational expressions.

Factoring is a powerful tool in algebra that streamlines computations and provides clarity in manipulation of expressions.

rational equations

Rational equations are equations involving rational expressions, making them crucial in algebraic problem-solving.
They may seem complex, but with the right approach, you can understand and solve them effectively.

• **Definition of Rational Equations**: These equations involve one or more rational expressions. A rational expression is any expression that can be written as \(\frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials, and \(q(x) eq 0\).
• **Solving Rational Equations**: The main strategy in solving them involves finding a common denominator to simplify the equation, then solving for the variable. This is often accomplished through:
  • **Clearing out Denominators**: Multiply through by the LCD to eliminate denominators. This converts a rational equation into a polynomial equation, which is typically easier to solve.
  • **Checking Solutions**: Because rational equations can introduce extraneous solutions, it's important to substitute back into the original equation to confirm solutions remain valid.
• **Application**: Rational equations model real-world situations where ratios and fractional parts are involved, making them practical in a wide range of fields.

By understanding rational equations, you effectively bridge the gap between pure algebraic manipulation and practical problem-solving.