Problem 6 - Algebra 1: Concepts and Skills

Question

Simplify the expression. If not possible, write already in simplest form. $$ \frac{12 x^{2}}{6 x} $$

Step-by-Step Solution

Answer

The simplified expression of \(\frac{12x^{2}}{6x}\) is \(2x\).

Step 1: Identify common factors

In this case, \(12x^{2}\) and \(6x\), it is evident that both expressions are divisible by \(6x\). Therefore, \(6x\) is the common term.

Step 2: Simplify the expression

To simplify the expression, divide both the numerator and denominator by the common term: \(\frac{12x^{2}}{6x} = \frac{12x^{2}/6x}{6x/6x} = \frac{2x}{1}\)

Step 3: Final Simplified Expression

Since we can't divide any further, the expression is in its simplest form. Writing the final expression without the denominator 1, we get \(2x\).

Key Concepts

Common Factors

Simplifying algebraic expressions often involves identifying common factors. A common factor is a number or expression that divides two or more numbers or expressions without leaving a remainder. In our exercise, the expressions in the numerator and denominator are \(12x^2\) and \(6x\), respectively.

Finding a common factor is just like sharing something evenly among a group. You want to find what can evenly divide both the top (numerator) and the bottom (denominator) of your fraction.

• Look at the numbers: 12 and 6. Both can be divided by 6 evenly.
• Look at the variables: \(x^2\) and \(x\). Both share \(x\) as a factor.

Combine those observations, and you find \(6x\) as the common factor. Recognizing this is crucial because factoring out common elements helps us simplify expressions efficiently.

Numerator and Denominator

In fractions, the numerator is the top part, and the denominator is the bottom part. These elements tell us how many parts we are looking at (numerator) out of how many total parts (denominator).

More specifically, examining the expression \(\frac{12x^2}{6x}\), the numerator is \(12x^2\), meaning we have 12 times the square of \(x\). The denominator is \(6x\), which gives context as six parts of an \(x\). To simplify, understanding how the numerator and denominator relate helps establish what can "cancel out" by dividing both parts by their greatest common factor.

Remember:

• The denominator cannot be zero, as division by zero is undefined.
• The process focuses on dividing both parts by exactly the same factor to keep the expression's value unchanged.

Recognizing how numerators and denominators relate in an expression is a foundational skill in algebra.

Simplest Form

Achieving the simplest form of an algebraic expression means reducing it as much as possible while keeping its value. When an expression is simplified, it is easier to understand and use for further calculations.

In our example, once you divide both the numerator \(12x^2\) and the denominator \(6x\) by the common factor \(6x\), you are left with \(\frac{2x}{1}\). This simplifies further to just \(2x\), eliminating the unnecessary "1" in the denominator.

• Simplification ensures there are no common factors left in the expression.
• This step reduces complexity in solving equations or inequalities involving the expression.
• It also provides clarity in communication. A simpler expression is straightforward, aiding comprehension and minimizing errors.

Thus, by following all simplification steps correctly, one finds the simplest form of the expression as clearly as possible.
Understanding this process is empowering for tackling more advanced algebraic problems, as it is a fundamental building block for resolving equations efficiently.