Problem 45 - Elementary Algebra

Question

For the following problems, perform the indicated operations. $$ \frac{x+6}{x-1} \cdot \frac{x+7}{x+6} $$

Step-by-Step Solution

Answer

Question: Multiply the following rational expressions and simplify: $$ \frac{x+6}{x-1} \cdot \frac{x+7}{x+6} $$ Answer: After multiplying and simplifying the given rational expressions, the result is $$ \frac{x+7}{x-1} $$.

Step 1: Identify the given expressions and operation

We are given the following rational expressions and the operation is multiplication: $$ \frac{x+6}{x-1} \cdot \frac{x+7}{x+6} $$

Step 2: Simplify the expressions

Before multiplying the expressions, let's check if we can simplify them. We notice that both the numerator of the second expression and the numerator of the first expression have the same term \((x+6)\). This means that we can cancel this term from both numerators. $$ \frac{(x+6)}{x-1} \cdot \frac{(x+7)}{(x+6)} = \frac{\cancel{(x+6)}}{x-1} \cdot \frac{(x+7)}{\cancel{(x+6)}} $$

Step 3: Perform the multiplication

Now that we have simplified the expressions, we can perform the multiplication: $$ \frac{1}{x-1} \cdot \frac{x+7}{1} $$ Multiply the numerators together and the denominators together: $$ \frac{1\cdot (x+7)}{(x-1)\cdot 1} = \frac{x+7}{x-1} $$ The result of the multiplication of the given expressions is: $$ \frac{x+7}{x-1} $$

Key Concepts

Multiplication of Rational Expressions

When dealing with rational expressions, multiplying them is similar to multiplying regular fractions. A rational expression is simply a ratio of two polynomials. In the exercise, we have two such expressions:

• \( \frac{x+6}{x-1} \)
• \( \frac{x+7}{x+6} \)

To multiply these expressions, you multiply the numerators together and then multiply the denominators together. This is done straightforwardly much like 'top times top' and 'bottom times bottom.' But before jumping into these calculations, it's a great idea to simplify them first. This makes solving easier and often prevents errors down the line.

Simplification

Simplification is key when working with rational expressions and fractions in general. The goal of simplification is to make complex expressions more manageable by reducing them to their simplest form. In the given exercise, we noticed that the numerator of the second expression, \( x+6 \), is the same as the numerator of the first expression. This allows us to cancel them out before multiplication. By canceling these terms, the expressions simplify to:\[\frac{\cancel{x+6}}{x-1} \cdot \frac{x+7}{\cancel{x+6}} = \frac{1}{x-1} \cdot \frac{x+7}{1}\]By reducing the expressions first, the multiplication becomes easier. The advantage of simplification is that it minimizes the work needed in subsequent steps and reduces chances of error.

Cancellation in Fractions

Cancellation in fractions and rational expressions is a very helpful process that can make calculations much simpler. It's essentially the removal of duplicate terms from both the numerator and the denominator. For example, if both have the common factor of \( x+6 \), we can "cancel it" as long as it is not in the context where it equals zero (since division by zero is undefined).In our exercise:

• The \((x+6)\) terms were present in both the numerator of the first expression and the denominator of the second expression.
• By canceling these terms, we simplified the problem considerably.

This cancellation leads directly to simplified expressions which make further calculations more straightforward, ultimately leading to: \[\frac{x+7}{x-1}\]Remember, though, always ensure that any common terms can be canceled legally, without affecting the mathematical truth of the expression, ensuring no zero divisions.