Question

Express as a single fraction $$\frac{3}{x-4}-\frac{2}{(x-4{)}^2}$$

Short answer

Answer: The simplified single fraction is $\frac{3x-14}{(x-4{)}^2}$.

Step-by-step solution

Identify the LCM of the denominators

The denominators are $(x-4)$ and $(x-4{)}^2$. Since the second denominator is just the square of the first one, the LCM will be $(x-4{)}^2$.

Express both fractions with the LCM as their denominator

In order to combine the fractions, we need to express both fractions using the LCM $(x-4{)}^2$ as their denominator. The first fraction already has $(x-4)$ in the denominator, so we need to multiply both the numerator and denominator by $(x-4)$: $$\frac{3}{x-4}\times \frac{x-4}{x-4}=\frac{3(x-4)}{(x-4{)}^2}$$ The second fraction already has the LCM as its denominator, so it remains unchanged: $$-\frac{2}{(x-4{)}^2}$$

Combine the fractions

Now that both fractions have the same denominator, we can combine them: $$\frac{3(x-4)}{(x-4{)}^2}-\frac{2}{(x-4{)}^2}$$

Simplify the resulting fraction

We can combine the numerators in the same fraction: $$\frac{3(x-4)-2}{(x-4{)}^2}$$ Now we can distribute the $3$ in the first term: $$\frac{3x-12-2}{(x-4{)}^2}$$ Finally, we can combine the constant terms: $$\frac{3x-14}{(x-4{)}^2}$$ The simplified single fraction is: $$\frac{3x-14}{(x-4{)}^2}$$

Key concepts

LCM (Least Common Multiple)

When dealing with algebraic fractions, finding the Least Common Multiple (LCM) of the denominators is essential. It allows you to combine fractions into a single, simple expression.

The LCM is the smallest expression that both denominators can divide without leaving a remainder. Look at the exercise: the denominators are $(x-4)$ and $(x-4{)}^2$.

• Since $(x-4{)}^2$ is simply the square of $(x-4)$, the LCM here will naturally be $(x-4{)}^2$.
• This is because $(x-4{)}^2$ encompasses both $(x-4)$ and its square, making it the smallest expression both can divide into.

Recognizing this helps simplify the process of working with fractions by providing a common base for all components.

Fraction Simplification

Simplifying fractions in this context means reducing fractions to their simplest form.

In the provided solution, the first step to simplification is transforming both fractions so they share the LCM $(x-4{)}^2$ as their denominator.

Here's how you proceed:

• Multiply the numerator and denominator of the first fraction, $\frac{3}{x-4}$, by $(x-4)$ to get $\frac{3(x-4)}{(x-4{)}^2}$.
• The second fraction, $-\frac{2}{(x-4{)}^2}$, already has the correct denominator.

Next, combine the fractions by adding or subtracting the numerators while keeping the denominator the same:

$\frac{3(x-4)-2}{(x-4{)}^2}$.

Distribute and combine like terms in the numerator to finish the simplification. This process is crucial in reducing complex expressions to a form that is easier to interpret and use.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. Mastering them helps in understanding how to manipulate and transform equations effectively.

In our example, variables and expressions like $(x-4)$ play a key role:

• Understand $3(x-4)$ as a multiplication between the constant $3$ and the expression $x-4$.
• When combining algebraic fractions, treat each variable-based expression as a unit in calculations.
• Distributing $3$ across $x-4$ involves expanding to $3x-12$, a standard algebraic manipulation.

Recognizing such patterns helps you confidently approach more complex algebraic expressions and aids in efficient problem-solving.