simplify (x+h) - (x) over h     x = square root of x and h is not equal to 0

To simplify the expression \(\frac{\sqrt{x+h} - \sqrt{x}}{h}\), let's proceed step by step:

1. **Start with the given expression:**

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

2. **Rationalize the numerator:**

   To simplify the expression, multiply both the numerator and denominator by the conjugate of the numerator. The conjugate of \(\sqrt{x+h} - \sqrt{x}\) is \(\sqrt{x+h} + \sqrt{x}\).

   \[
   \frac{\sqrt{x+h} - \sqrt{x}}{h} \cdot \frac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h} + \sqrt{x}} = \frac{(\sqrt{x+h} - \sqrt{x})(\sqrt{x+h} + \sqrt{x})}{h(\sqrt{x+h} + \sqrt{x})}
   \]

3. **Simplify the numerator using the difference of squares formula:**

   The numerator becomes:

   \[
   (\sqrt{x+h})^2 - (\sqrt{x})^2 = (x+h) - x = h
   \]

   So the expression now simplifies to:

   \[
   \frac{h}{h(\sqrt{x+h} + \sqrt{x})}
   \]

4. **Cancel out the \(h\) terms:**

   Since \(h\) is not equal to 0, you can cancel \(h\) from the numerator and the denominator:

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

Thus, the simplified form of the expression is:

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

Would you like further details or have any other questions?

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Here are 5 related questions:

1. How do you rationalize the denominator in an expression?
2. What is the difference of squares, and how is it applied in simplifications?
3. How can you apply this simplification to a specific value of \(x\) and \(h\)?
4. What are common techniques for simplifying radical expressions?
5. How does the limit of this expression behave as \(h\) approaches 0?

**Tip:** When simplifying expressions involving radicals, rationalizing the numerator or denominator often makes the process easier and more straightforward.

Learn how to simplify the radical expression (sqrt(x+h) - sqrt(x)) / h step-by-step. This problem involves algebraic manipulation of radical expressions and understanding limit theory. Suitable for advanced high school students.

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