Problem 38 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{18(x+2)^{3}}{16(x+2)^{2}}$$

Step-by-Step Solution

Answer

\(\frac{9x + 18}{8}\)

Step 1: Identify and Write the Fraction

The given algebraic fraction is \( \frac{18(x+2)^{3}}{16(x+2)^{2}} \).

Step 2: Factor the Numerator and Denominator

Notice that both the numerator and the denominator have the factor \((x+2)\). Write the problem as \( \frac{18(x+2) \cdot (x+2)^2}{16 \cdot (x+2)^2} \).

Step 3: Cancel Common Factors

Since \((x+2)^2\) is a common factor in both the numerator and the denominator, cancel it out: \( \frac{18(x+2)}{16} \).

Step 4: Simplify Further

Now, focus on simplifying the numerical coefficient: \( \frac{18}{16} \). This reduces to \( \frac{9}{8} \). So the fraction becomes \( \frac{9(x+2)}{8} \).

Step 5: Write the Final Simplified Expression

Distribute the term in the numerator as needed: \( \frac{9(x+2)}{8} = \frac{9x + 18}{8} \).

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form while keeping their value the same. For algebraic fractions, this means getting rid of any complicated expressions that are repeated in both the numerator and the denominator.

Here's why simplification is helpful:

• It makes the fraction easier to work with in subsequent steps of a mathematical problem.
• It helps you see the "core" behavior of the function or expression represented by the fraction.

In algebraic fractions, start by checking if there are any common factors in the numerator and the denominator that can be simplified. For example, if the numerator and the denominator contain a similar algebraic expression, it can be factored out to simplify the fraction further.

Factoring

Factoring is the process of splitting an expression into a product of simpler entities, or factors, that when multiplied together give you the original expression. This step is crucial when simplifying algebraic fractions as it allows for the identification of common factors.

To factor both parts of the expression, look for common terms:

• In the numerator, see if there's a term or expression that can be written as a product.
• Similarly, check the denominator for common factors.

For example, in our problem, notice that both the numerator and denominator have \( (x+2)^2 \) as a factor. Writing these larger expressions as simpler products aids in canceling common factors later.

Canceling Common Factors

Canceling common factors is a powerful technique in simplifying fractions. Once both the numerator and the denominator are factored, check for factors that appear in both.

Steps to cancel common factors:

• Identify terms or expressions that are identical in both the numerator and the denominator.
• "Cancel out" these common factors because dividing a term by itself equals 1, which simplifies the expression.

For our exercise, cancel \( (x+2)^2 \) from both to leave us with \( \frac{18(x+2)}{16} \). This simplification step is key to achieving the more manageable \( \frac{9(x+2)}{8} \), simplifying further to the final expression of \( \frac{9x+18}{8} \).