Problem 26 - Algebra 1

Question

Simplify the expression. $$\frac{8}{2+3 x} \cdot(8+12 x)$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{96x + 64}{2+3x} \)

Step 1: Distribute the Multiplication

Firstly, distribute the fraction \( \frac{8}{2+3x} \) with each term in the parenthesis \( (8+12x) \). This results in: \( \frac{8}{2+3x} \cdot 8 + \frac{8}{2+3x} \cdot 12x \)

Step 2: Simplify the Expressions

After doing the multiplication, simplify each fraction. The first becomes \( \frac{64}{2+3x} \) and the second one becomes \( \frac{96x}{2+3x}\)

Step 3: Convert to Common Denominators

Both expressions have the same denominator \(2+3x\). This could be written as: \( \frac{64 + 96x}{2+3x} \)

Step 4: Final Simplification

Rearrange the numerator to get the final simplified expression. The solution is: \( \frac{96x + 64}{2+3x} \)

Key Concepts

Simplification

Simplification is a process in mathematics where we try to make expressions easier to understand and work with. When simplifying, the goal is to reduce an expression to its simplest form without changing its value. This is very useful, especially in algebra, where expressions can get quite complicated.

• To simplify, always look for common factors and see if you can cancel them out.
• Combine like terms if possible and try to reduce fractions to their simplest form.
• In rational expressions, we simplify by dividing both the numerator and the denominator by their greatest common factor, if applicable.

Simplification makes calculations more straightforward and helps in solving equations more easily. In the exercise provided, we simplified the initial expression through distribution and then reduced the expression by combining like terms.

Distributive Property

The distributive property is a fundamental algebraic rule that helps in simplifying expressions. It states that multiplying a number by a sum gives the same result as doing each multiplication separately.
In formula terms, the distributive property is:
\( a(b + c) = ab + ac \).
For the given exercise, we distributed the fraction \( \frac{8}{2+3x} \) to each term inside the parenthesis \( (8 + 12x) \).

• This means that each term in the parenthesis gets multiplied by \( \frac{8}{2+3x} \).
• This results in two separate terms: \( \frac{8}{2+3x} \cdot 8 \) and \( \frac{8}{2+3x} \cdot 12x \).

Using the distributive property correctly ensures that all parts of the expression are accounted for in calculations.
It simplifies the process considerably when dealing with algebraic fractions.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both are algebraic expressions.
These types of fractions can look complicated, but they follow the same rules as regular fractions. The goal is to simplify them as much as possible while retaining their value. Here are some steps to consider:

• To simplify an algebraic fraction, you can often factor the numerator and the denominator and then cancel out common factors.
• Make sure that you do not cancel terms before thoroughly checking for common factors.
• Always keep the values in the simplest form, and pay attention to any restrictions that may arise (like values that make the denominator zero).

In the exercise we were given, the numerator was simplified after using the distributive property, resulting in a simplified algebraic fraction.
All components worked together to solve the expression fully.