Problem 85 - Algebra 1: Concepts and Skills

Question

Write the product in simplest form. $$\frac{x}{x+6} \cdot \frac{x+6}{x+1}$$

Step-by-Step Solution

Answer

The product in simplest form is \(\frac{x}{x+1}\).

Step 1: Rewrite the product

Start by writing down the product exactly as it is:\[\frac{x}{x+6} \cdot \frac{x+6}{x+1}\]

Step 2: Simplify the Product

Notice that the denominator of the first fraction and the numerator of the second fraction are identical. Because of this, you can simplify the product by canceling out these identical terms: \[\frac{x}{\cancel{x+6}} \cdot \frac{\cancel{x+6}}{x+1}\]This leaves you with\[\frac{x}{x+1}\]

Key Concepts

Cancellation in Algebraic Fractions

Understanding how to simplify algebraic fractions through cancellation is a key skill in algebra. Cancellation involves reducing fractions by eliminating factors that appear in both the numerator and the denominator.

Consider an algebraic fraction like \(\frac{x}{x+6} \cdot \frac{x+6}{x+1}\). You can cancel terms when they are the same in the numerator of one fraction and the denominator of another. Here, \(x+6\) appears on top in one fraction and on the bottom in the other. This shared factor can be cancelled out, resulting in a simpler expression, \(\frac{x}{x+1}\).

Cancellation does not change the value of the fraction, but it does make the expression more straightforward to work with. Always ensure that you cancel only factors (expressions that are multiplied together) and not terms that are added or subtracted. This helps avoid common mistakes and paves the way towards mastering algebraic fraction manipulation.

Simplifying Complex Fractions

Complex fractions are fractions that contain a fraction in the numerator, denominator, or both. Simplifying them might seem daunting, but a good strategy can make the process manageable.

Start by identifying any common factors between the numerators and denominators across the complex fraction. If there's a factor that exactly matches in one numerator and one denominator, as in our example \(\frac{x}{x+6} \cdot \frac{x+6}{x+1}\), you can simplify by applying cancellation, as mentioned earlier.

Step-by-Step Simplification

Apply cancellation (when safe to do so), next, combine fractions if necessary, and finally, simplify the remaining expression. When following these steps, also look out for opportunities to factor expressions further, as this can sometimes reveal additional cancellation possibilities. Comparable to reducing to lowest terms in simple fractions, simplifying complex fractions makes them easier to evaluate and use in future calculations.

Algebraic Expression Manipulation

Algebraic expression manipulation involves rearranging and simplifying expressions without changing their overall value. This skill is vital when working with algebraic fractions.

Key manipulation techniques include expanding (using the distributive property), factoring (the reverse of expanding), and cancelling shared factors in fractions. By mastering these techniques, you can transform complex algebraic expressions into simpler, more manageable forms — much like how \(\frac{x}{x+6} \cdot \frac{x+6}{x+1}\) was simplified to \(\frac{x}{x+1}\).

Practice Makes Perfect

Consistent practice with these techniques builds strong algebraic intuition. When encountering algebraic fractions, always consider which manipulation can simplify the expression. Over time, identifying common factors, terms that can be cancelled, and opportunities to factor or expand becomes second nature. This proficiency turns complex fraction simplification into a straightforward task.