\frac{1}{\sqrt{x + h}} - \frac{1}{\sqrt{x}}

The given expression is:

\[
\frac{1}{\sqrt{x + h}} - \frac{1}{\sqrt{x}}
\]

To simplify or manipulate this, we can combine the two terms by finding a common denominator. The least common denominator (LCD) between the two fractions is \(\sqrt{x + h} \cdot \sqrt{x}\). Here's the step-by-step process:

### Step 1: Multiply each fraction by the necessary factors
\[
\frac{1}{\sqrt{x + h}} \cdot \frac{\sqrt{x}}{\sqrt{x}} - \frac{1}{\sqrt{x}} \cdot \frac{\sqrt{x + h}}{\sqrt{x + h}}
\]

This gives us:
\[
\frac{\sqrt{x}}{\sqrt{x + h} \cdot \sqrt{x}} - \frac{\sqrt{x + h}}{\sqrt{x + h} \cdot \sqrt{x}}
\]

### Step 2: Combine the two fractions
Since the denominators are now the same, we can write:
\[
\frac{\sqrt{x} - \sqrt{x + h}}{\sqrt{x + h} \cdot \sqrt{x}}
\]

This is the simplified form of the given expression.

Would you like further details or have any questions?

Here are 5 related questions:
1. How would this expression behave as \(h\) approaches 0?
2. Can you evaluate this expression for specific values of \(x\) and \(h\)?
3. How would you find the derivative of this expression with respect to \(x\)?
4. How does this expression relate to the difference quotient for square roots?
5. Can this expression be used in limits involving small \(h\) values?

**Tip**: When simplifying fractions, always check if a common denominator will help in combining terms efficiently.

Learn how to simplify the expression \frac{1}{\sqrt{x + h}} - \frac{1}{\sqrt{x}} by finding a common denominator and combining the fractions. This solution also explores related concepts such as the difference quotient and limits, suitable for students in Grades 10-12.

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