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The first problem asks you to simplify the following fractional expression:

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

### Step-by-step simplification:
1. Combine the terms in the numerator:
   \[
   \frac{1}{\sqrt{x + h}} - \frac{1}{\sqrt{x}} = \frac{\sqrt{x} - \sqrt{x + h}}{\sqrt{x} \cdot \sqrt{x + h}}
   \]
   To do this, find a common denominator for the two terms in the numerator.

2. The expression now becomes:
   \[
   \frac{\frac{\sqrt{x} - \sqrt{x + h}}{\sqrt{x(x + h)}}}{h}
   \]

3. Simplify the fraction by multiplying the numerator and denominator:
   \[
   = \frac{\sqrt{x} - \sqrt{x + h}}{h \cdot \sqrt{x(x + h)}}
   \]

This is the simplified form of the given expression.

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Now for the second problem, we are asked to simplify the following expression:

\[
\frac{8(2 + x)^{1/3} - x(2 + x)^{-2/3}}{(2 + x)^{2/3}}
\]

### Simplification:
1. Factor out \( (2 + x)^{-2/3} \) from the numerator:
   \[
   \frac{(2 + x)^{-2/3} \left(8(2 + x) - x\right)}{(2 + x)^{2/3}}
   \]

2. Simplify the powers:
   \[
   = (2 + x)^{-2/3 + 2/3} \left(8(2 + x) - x\right) = \left(8(2 + x) - x\right)
   \]

3. Expand the terms:
   \[
   = 16 + 7x
   \]

Thus, the simplified form of the expression is \( 16 + 7x \).

Would you like any further clarification or details on any of the steps?

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Here are five related questions to expand your understanding:
1. How do you handle expressions with nested fractions more efficiently?
2. Can you explain the general approach to simplifying radical expressions?
3. What happens to the expression as \( h \to 0 \) in the first problem (link to derivatives)?
4. How can we simplify expressions that involve powers and roots simultaneously?
5. What role does factoring play in the simplification process of expressions with variables in both the base and the exponent?

**Tip:** Always try to find a common denominator when dealing with complex fractions!

Simplify the fractional expression:
(1/√(x + h) - 1/√x) / h
and simplify the second expression:
(8(2 + x)^(1/3) - x(2 + x)^(-2/3)) / (2 + x)^(2/3).

Learn how to simplify fractional expressions involving radicals and exponents, including (1/√(x + h) - 1/√x) / h and (8(2 + x)^(1/3) - x(2 + x)^(-2/3)) / (2 + x)^(2/3). Detailed step-by-step simplifications included.

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