Problem 14 - Algebra 2 and Trigonometry

Question

In \(13-24,\) divide and express each quotient in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{12}{a} \div \frac{6}{4 a} $$

Step-by-Step Solution

Answer

The simplest form is 8, and \(a \neq 0\) for the fractions to be defined.

Step 1: Write the Division as Multiplication by the Reciprocal

The given division operation involves fractions: \( \frac{12}{a} \div \frac{6}{4a} \). To divide by a fraction, multiply by its reciprocal:\[ \frac{12}{a} \times \frac{4a}{6} \]

Step 2: Simplify the Expression

Now simplify the expression step-by-step. First, multiply the numerators and the denominators:\[ \frac{12 \times 4a}{a \times 6} \] Next, cancel out common factors in the numerator and the denominator. The \(a\) in the numerator and the denominator cancel each other out, and the 6 in 12 and 6 also can be simplified:\[ \frac{2 \times 4}{1} = \frac{8}{1} = 8 \]

Step 3: Identify Values that Make the Original Fractions Undefined

In the original fractions \(\frac{12}{a}\) and \(\frac{6}{4a}\), the variable \(a\) is in the denominator, so the fractions are undefined when \(a = 0\). Therefore, \(a\) cannot be 0.

Key Concepts

Simplification of Expressions

When we talk about the simplification of expressions, we're aiming to make mathematical expressions as simple as possible. In the context of polynomial division, we often deal with complex fractions.

• Start by rewriting the division problem as a multiplication problem, using the reciprocal.
• Multiply both fractions together, and simplify where you can by canceling out common terms.

For example, in the expression \(\frac{12}{a} \div \frac{6}{4a}\), you can simplify it by multiplying \(\frac{12}{a}\) by the reciprocal of \(\frac{6}{4a}\), which is \(\frac{4a}{6}\). Then, look for terms that appear in both the numerator and the denominator.
By doing this, you will notice that the variable \(a\) and the number 6 are common in both, allowing them to be canceled out. This step drastically simplifies the fraction to \(\frac{8}{1}\) or just \(8\). The simplification of expressions can make problems easier to work with and understand.

Reciprocals in Division

Reciprocals are a fundamental part of dividing fractions. To find the reciprocal of a fraction, simply flip the numerator and the denominator.

• This switch allows us to transform a division operation into a multiplication one.
• Using reciprocals is a handy tool because multiplication is often easier to manage.

In our example of \(\frac{12}{a} \div \frac{6}{4a}\), we convert the division problem into a multiplication problem by taking the reciprocal of \(\frac{6}{4a}\). Therefore, it becomes \(\frac{12}{a} \times \frac{4a}{6}\). By applying reciprocals, the division of fractions becomes a more straightforward process, allowing for an easier simplification.
Remember: the reciprocal process is all about flipping! Understanding this is key to mastering polynomial division with fractions.

Undefined Fractions

Undefined fractions occur when the denominator of a fraction is zero. In mathematics, division by zero is not allowed because it leads to an undefined or indeterminate situation.

• Fractions are undefined at any value that makes their denominator zero.
• Identifying these values is crucial for understanding the domain of a fraction.

Going back to our example, both fractions in the original expression, \(\frac{12}{a}\) and \(\frac{6}{4a}\), have \(a\) in the denominator. If \(a = 0\), both fractions become undefined because you cannot have zero in the denominator of a fraction.
Therefore, values that make a denominator zero are critical in determining where a fraction is undefined. Observing the denominators and pinpointing these critical values ensure you avoid undefined fractions, maintaining the validity of your calculations.