Problem 6 - 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. $$ \frac{3 a}{5} \cdot \frac{10}{9 a} $$

Step-by-Step Solution

Answer

The simplest form is \( \frac{2}{3} \) and it is undefined for \( a = 0 \).

Step 1: Identify the expression

We need to multiply the two fractions \( \frac{3a}{5} \) and \( \frac{10}{9a} \).

Step 2: Multiply the numerators and denominators

To multiply two fractions, multiply the numerators together and the denominators together: \\( \frac{3a}{5} \times \frac{10}{9a} = \frac{3a \cdot 10}{5 \cdot 9a} \).

Step 3: Simplify the expression

Simplify \( \frac{3a \cdot 10}{5 \cdot 9a} \) by canceling out the common factor \( a \), if \( a eq 0 \): \\( \frac{30a}{45a} = \frac{30}{45} \). \Further simplify \( \frac{30}{45} \) by dividing both the numerator and the denominator by 15: \\( \frac{30 ÷ 15}{45 ÷ 15} = \frac{2}{3} \).

Step 4: Determine undefined values

The original fractions are undefined when the denominator equals zero, which occurs at \( a = 0 \). Therefore, the fraction is not defined for \( a = 0 \).

Key Concepts

Simplifying Fractions

When multiplying fractions, it's important to simplify wherever possible. This process involves reducing fractions to their simplest form so that you can see the most straightforward representation of the numbers. In the given exercise, we have two fractions: \( \frac{3a}{5} \) and \( \frac{10}{9a} \). To simplify after multiplication, you first multiply the numerators (top numbers) and denominators (bottom numbers) together. This gives you \( \frac{3a \cdot 10}{5 \cdot 9a} \).

Next, we simplify by canceling out common terms. Here, the term \( a \) appears both in the numerator and the denominator, so if \( a eq 0 \), it can be canceled out, resulting in \( \frac{30}{45} \). This fraction can further be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 15 in this case, leaving you with \( \frac{2}{3} \).

Remember, the process of simplifying ensures clarity and accuracy in mathematical calculations, making it crucial in algebraic operations.

Undefined Values

An important consideration when dealing with fractions, especially in algebra, is identifying when a fraction becomes undefined. A fraction is undefined when its denominator is zero, because division by zero is not possible in mathematics.

In the provided exercise, the division by zero occurs when \( a = 0 \). This is because substituting 0 for \( a \) makes the denominator of the fraction zero, which breaks the rules of arithmetic. Thus, \( a \) cannot equal zero in this context. It's essential to note such restrictions on variable values to avoid undefined expressions. Always remember to solve for any variable values causing the denominator of fractions to equal zero to ensure the mathematical expression remains valid.

Algebraic Fractions

Algebraic fractions are essentially ratios of two algebraic expressions. They can be manipulated very much like numeric fractions, with additional considerations for any variables involved. In algebraic fractions, like the ones in this exercise, variables are a key factor in operations like multiplication and simplification.

To successfully work through algebraic fractions, first treat the variables as part of the numbers you are working with. Identify common factors in both the numerators and denominators. For instance, in the fraction \( \frac{3a}{5} \times \frac{10}{9a} \), the \( a \) in both the numerator and denominator serve as common factors that can be canceled out, making simplification possible.

Handling algebraic fractions requires precision, as variables can introduce additional limitations on what values are permissible. This crucial understanding allows the fractions to be simplified correctly while respecting mathematical rules. Always remember to simplify whenever possible to reveal the most basic form of the expression.