It looks like you've uploaded a page with a series of mathematical problems involving square roots and other algebraic expressions. I’ll go through and solve each one individually, providing the details for each calculation.

Let's begin:

1. Problem (a):
   $|a - \sqrt{a}| - |a - \sqrt{a}| + |a - \sqrt{a}|$ This expression evaluates based on the value of $a$. For certain values of $a$, the expression inside the absolute values might be positive or negative. I'll need more specifics to calculate a general simplification.

2. Problem (b):
   $\sqrt{\frac{-27}{216}}$ This expression simplifies as follows:

   • First, simplify $\frac{-27}{216} = -\frac{1}{8}$.
   • Since we are taking the square root of a negative number, this will yield an imaginary number: $\sqrt{-\frac{1}{8}} = \frac{i}{2\sqrt{2}}$

3. Problem (c):
   $\sqrt{\dots16}$
   The notation here is unclear; it appears there may be some values missing or intended for the reader to fill in.

4. Problem (d):
   $\sqrt{4.96} \approx 2.228$

5. Problem (e):
   $\sqrt{\frac{128}{2187}}$
   Simplify as follows:

   • $\frac{128}{2187}$ does not simplify nicely, but we can approximate the square root: $\sqrt{\frac{128}{2187}} \approx 0.2415$

6. Problem (f):
   $\frac{\sqrt{0.04} + \sqrt{0.08}}{\sqrt{0.16} - \sqrt{0.0016}}$

   • Calculate each square root individually: $\sqrt{0.04} = 0.2, \quad \sqrt{0.08} \approx 0.2828$ $\sqrt{0.16} = 0.4, \quad \sqrt{0.0016} = 0.04$
   • Substitute these values in: $\frac{0.2 + 0.2828}{0.4 - 0.04} = \frac{0.4828}{0.36} \approx 1.3411$

7. Problem (g):
   $\sqrt[3]{\dots32 \times \sqrt[3]{0.16^{-1}}}$ This problem appears to be missing some elements or specific values.

8. Problem (h):
   $\sqrt[3]{16^{\frac{5}{3}}}$ Simplify as follows: $\sqrt[3]{16^{\frac{5}{3}}} = 16^{\frac{5}{9}}$

9. Problem (i):
   $\sqrt{\left( (x^{-1})^{-5} - (y^{-1})^{7} \right)}$

   • Simplify each component: $(x^{-1})^{-5} = x^5, \quad (y^{-1})^{7} = y^{-7}$ $\sqrt{x^5 - y^{-7}}$

10. Problem (j):
    This expression contains multiple square roots with unclear values. I need clarification on specific variables or values.

11. Problem (k):
    $\frac{(\sqrt{a^b \cdot b^a})^f}{\sqrt{a^a \cdot b^b}}$ Simplify each square root and apply the power rules.

Would you like detailed solutions for specific problems, or do you need clarification on the interpretation of any problems?

Follow-up Questions:

1. What are the specific values for any missing variables in the problems?
2. Would you like to focus on simplifying specific types of expressions, like those involving imaginary numbers?
3. Are there any particular steps you would like further elaboration on?
4. Do you need assistance with the general rules for simplifying square roots and cube roots?
5. Would you like to go over properties of absolute values in algebraic expressions?

Tip:

When simplifying expressions with roots and fractions, always try to reduce fractions and exponents before applying roots to make calculations easier.