Chapter 5: Problem 9

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

Short Answer

Expert verified

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: $$ \frac{3x-14}{(x-4)^2} $$

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

Short Answer

Step by step solution

Identify the LCM of the denominators

Express both fractions with the LCM as their denominator

Combine the fractions

Simplify the resulting fraction

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