Problem 12 - Algebra 2 and Trigonometry

Question

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{5 a^{2} b}{10 a}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{ab}{2}\), and it is undefined for \(a = 0\).

Step 1: Identify Like Terms

First, look at the numerator and denominator to identify any like terms that can be canceled. In the expression \(\frac{5a^2b}{10a}\), the term \(a\) appears in both the numerator and the denominator.

Step 2: Factor the Coefficients

Next, factor the coefficients in both the numerator and the denominator. The coefficient in the numerator is 5 and in the denominator is 10. Notice that 5 is a factor of both numbers.

Step 3: Simplify the Coefficients

Divide the coefficients by their greatest common divisor. Since 5 is a factor, divide both 5 and 10 by 5, resulting in \(\frac{1}{2}\).

Step 4: Cancel the Common Variable

Cancel the common term \(a\) from both the numerator and the denominator. Since \(a^2\) is in the numerator, and \(a\) is in the denominator, this simplifies: \(\frac{a}{a} = 1\). You are left with \(a\) in the numerator.

Step 5: Write the Simplified Expression

Combine the results from Step 3 and Step 4 to write the simplified expression: \(\frac{ab}{2}\).

Step 6: Determine Undefined Values

The fraction is undefined for values that make the denominator zero. Since the original denominator was \(10a\), the expression is undefined when \(a = 0\).

Key Concepts

Simplification

Simplifying rational expressions is similar to simplifying fractions in arithmetic. The idea is to write the expression in the most reduced form possible.
The goal is to make the expression simpler without changing its value. When we simplify

• Check for any common factors in the numerator and the denominator.
• Factor both the numerator and the denominator completely, if possible.
• Divide both by their greatest common factor (GCF) to reduce the expression.

For \[\frac{5a^2b}{10a}\]we identified "5a" as the common factor. Dividing the entire expression by this common factor gives us the simplest form: \[\frac{ab}{2}\]This process makes the expression easier to work with while maintaining equivalence.

Undefined Values

Rational expressions can sometimes have values at which they are undefined. These are points where the denominator of the fraction becomes zero because division by zero is undefined.Finding undefined values requires focusing on the denominator's variable.
Determine the value(s) of the variable which make the denominator zero.In the example \[\frac{5a^2b}{10a}\]we looked at the denominator, which is "10a".
The expression is undefined when\[10a = 0\]Solving for "a" gives us \[a = 0\]So, the expression is undefined when \(a = 0\). Always pay particular attention to these values, as they can indicate restrictions on the domain of the expression.

Variables

Variables represent unknown values or values that can change. They are used to generalize mathematical expressions, allowing them to represent real-world scenarios.
In rational expressions, variables can appear in both the numerator and denominator.
Their role is crucial as they define the behavior and restrictions of the expression.For \[\frac{5a^2b}{10a}\]we are dealing with variables "a" and "b". Understanding these symbols as placeholders helps in manipulating and solving expressions correctly.
They allow expressions to be adapted to different conditions and input values.
Remember to check how variables impact simplification and undefined values.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest number that can evenly divide two or more integers. Finding the GCD is a crucial part of simplifying rational expressions.For example, in the expression\[\frac{5a^2b}{10a}\]we need to factor the coefficients 5 and 10 to simplify.
The GCD of 5 and 10 is 5, as 5 divides both numbers without leaving a remainder.Simultaneously finding the GCD of variables (like \(a\)) involves looking at the powers. When we cancel out the lower powers, such as simplifying \(\frac{a^2}{a}\), we use the GCD of the powers, which in this case is \(a^1\).Using the GCD simplifies the expression efficiently and maintains its balance.
This technique is especially useful in algebra for streamlining and solving complex rational equations.