https://file.aiquickdraw.com/mathbot/file/1725845007303bph9jcig.jpg

Let's simplify the given expression:

\[
\frac{\frac{x - \frac{1}{x}}{\frac{5}{x} + \frac{5}{x^2}}}
\]

### Step 1: Simplify the numerator
The numerator is:

\[
x - \frac{1}{x}
\]

### Step 2: Simplify the denominator
The denominator is:

\[
\frac{5}{x} + \frac{5}{x^2}
\]

To simplify this, find a common denominator:

\[
\frac{5x + 5}{x^2} = \frac{5(x + 1)}{x^2}
\]

### Step 3: Write the expression as a division of two fractions
Now, the entire expression is:

\[
\frac{x - \frac{1}{x}}{\frac{5(x + 1)}{x^2}}
\]

This can be simplified by multiplying the numerator by the reciprocal of the denominator:

\[
\left( x - \frac{1}{x} \right) \times \frac{x^2}{5(x + 1)}
\]

### Step 4: Distribute and simplify
Distribute \( x^2 \) in the numerator:

\[
\frac{x^3 - x}{5(x + 1)}
\]

So, the fully simplified expression is:

\[
\frac{x(x^2 - 1)}{5(x + 1)}
\]

### Step 5: Factorize further if possible
Notice that \( x^2 - 1 \) can be factored as \( (x - 1)(x + 1) \):

\[
\frac{x(x - 1)(x + 1)}{5(x + 1)}
\]

Since \( x + 1 \) appears in both the numerator and denominator, you can cancel it out (assuming \( x \neq -1 \)):

\[
\frac{x(x - 1)}{5}
\]

This is the final simplified expression.

Would you like more details or have any questions?

---

**Related Questions:**

1. What is the next step if \( x + 1 \) could not be canceled?
2. How do you handle simplifications involving negative exponents?
3. What happens if \( x \) is a negative value in this expression?
4. How can we simplify more complex rational expressions?
5. What is the role of factoring in simplifying expressions like this one?

**Tip:** Always check if terms can be factored before simplifying, as it can make the process easier and more accurate.

Fully simplify the expression (x - 1/x) / (5/x + 5/x^2).

Learn how to fully simplify the rational expression (x - 1/x) / (5/x + 5/x^2) with a step-by-step solution. This problem covers key algebraic concepts, including simplifying rational expressions, finding a common denominator, and factoring. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation