Simplify the expression: \frac{1}{x} + \frac{3}{x^3}
Understand the Problem
The question asks to simplify the expression \frac{1}{x} + \frac{3}{x^3}. This involves finding a common denominator and combining the fractions.
Answer
$\frac{x^2 + 3}{x^3}$
Answer for screen readers
$\frac{x^2 + 3}{x^3}$
Steps to Solve
- Find the Least Common Denominator (LCD)
The denominators are $x$ and $x^3$. The LCD is $x^3$.
- Rewrite each fraction with the LCD
Rewrite $\frac{1}{x}$ with the denominator $x^3$. To do this, multiply the numerator and denominator by $x^2$:
$$ \frac{1}{x} = \frac{1 \cdot x^2}{x \cdot x^2} = \frac{x^2}{x^3} $$
The second fraction, $\frac{3}{x^3}$, already has the LCD as its denominator, so it remains as is.
- Add the fractions
Now that both fractions have the same denominator, we can add them:
$$ \frac{x^2}{x^3} + \frac{3}{x^3} = \frac{x^2 + 3}{x^3} $$
- Simplify the result
The expression $\frac{x^2 + 3}{x^3}$ is already simplified, as there are no common factors between the numerator and the denominator.
$\frac{x^2 + 3}{x^3}$
More Information
The simplified expression is a single fraction. The numerator is a quadratic expression, and the denominator is a cubic expression.
Tips
A common mistake is to try to cancel terms incorrectly. For example, one might try to cancel an $x^2$ term from the numerator and denominator, but this is not allowed because $x^2$ is not a factor of the entire numerator.
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