Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-12,\) multiply and express each product in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{5}{7 a} \cdot \frac{3 a}{20} $$

Step-by-Step Solution

Answer

The simplest form is \(\frac{3}{28}\), and the expression is undefined for \(a = 0\).

Step 1: Multiply the Numerators

To begin with, multiply the numerators of the fractions. The numerators are 5 and 3a. Thus, 5 multiplied by 3a gives \(5 \times 3a = 15a\).

Step 2: Multiply the Denominators

Next, multiply the denominators of the fractions. The denominators are 7a and 20. Therefore, 7a multiplied by 20 gives \(7a \times 20 = 140a\).

Step 3: Form the New Fraction

Combine the results from steps 1 and 2 to form the new fraction. This gives us \(\frac{15a}{140a}\).

Step 4: Simplify the Fraction

Cancel out the common factor \(a\) in the numerator and denominator, reducing the fraction to \(\frac{15}{140}\). Next, find the greatest common divisor of 15 and 140, which is 5, and divide both by 5 to simplify: \(\frac{15 \div 5}{140 \div 5} = \frac{3}{28}\).

Step 5: Determine Undefined Values

Note that the original fractions are undefined when the denominators are zero. Therefore, \(7a = 0\) or \(a = 0\) for the \(\frac{5}{7a}\). Thus, the expression is undefined when \(a = 0\).

Key Concepts

Undefined Expressions

Rational expressions can sometimes be undefined. This occurs when any denominator in the expression equals zero. For instance, in our original exercise, the denominator of the first fraction is \( 7a \). If \( a \) equals zero, then the expression \( \frac{5}{7a} \) becomes undefined because division by zero is impossible in mathematics. It is important to always check each variable in the denominator to see if there is a value that could make the denominator zero. If such a value exists, it is crucial to list it as a value for which the expression is not defined. To avoid encountering undefined values in rational expressions, make sure to always check the denominator of every fraction involved in the expression. If at any point the denominator becomes zero, make note that this specific value for the variable makes the expression undefined.

Multiplying Fractions

When multiplying fractions, the process is straightforward.

• Start by multiplying the numerators together.
• Then multiply the denominators together.

For example, in the exercise, we had fractions \( \frac{5}{7a} \) and \( \frac{3a}{20} \). First, we multiplied the numerators \( 5 \times 3a = 15a \). Next, we multiplied the denominators \( 7a \times 20 = 140a \). After multiplying the numerators and the denominators, form a new fraction. This gives us \( \frac{15a}{140a} \).Remember, the key steps:

• Multiply numerators together.
• Multiply denominators together.
• Form the resulting fraction from the products.

Always double-check your multiplication to ensure accuracy. Once complete, proceed to simplifying the expression.

Rational Expressions

A rational expression is essentially a fraction where both the numerator and the denominator are polynomials. Working with rational expressions involves operations similar to number fractions, but with additional steps due to variable presence. In the exercise, \( \frac{5}{7a} \) and \( \frac{3a}{20} \) are both rational expressions because they contain variables in their denominators and numerators. To simplify a rational expression:

• Perform any given operation, such as multiplication or addition.
• Cancel out common factors if they exist.
• Always ensure that the simplified expression still holds true to any restrictions on variable values to prevent undefined situations.

Upon multiplying, we obtained \( \frac{15a}{140a} \). We simplified this by removing the common factor \( a \), leading to \( \frac{15}{140} \). Further simplification involved finding the greatest common factor of 15 and 140, which is 5, resulting in \( \frac{3}{28} \). Simplifying rational expressions requires a careful attention to detail to ensure that no steps are overlooked and that the expression remains valid.