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The given mathematical expression is:

\[
\left(\frac{1}{9}\right)^{-\frac{1}{2}} + 4^2 \cdot \sqrt[3]{\frac{1}{64}} \times (25)^{-\frac{1}{2}} + 27^{\frac{1}{3}} \cdot (8)^{-\frac{1}{3}} \div \left( \sqrt{\frac{4}{9}} - 4^0 \right)
\]

Let's simplify this step by step:

### 1. Simplify \(\left(\frac{1}{9}\right)^{-\frac{1}{2}}\)
\[
\left(\frac{1}{9}\right)^{-\frac{1}{2}} = 9^{\frac{1}{2}} = \sqrt{9} = 3
\]

### 2. Simplify \(4^2 \cdot \sqrt[3]{\frac{1}{64}} \times (25)^{-\frac{1}{2}}\)
- \(4^2 = 16\)
- \(\sqrt[3]{\frac{1}{64}} = \frac{1}{\sqrt[3]{64}} = \frac{1}{4}\)
- \((25)^{-\frac{1}{2}} = \frac{1}{\sqrt{25}} = \frac{1}{5}\)

So:
\[
16 \cdot \frac{1}{4} \cdot \frac{1}{5} = \frac{16}{20} = \frac{4}{5}
\]

### 3. Simplify \(27^{\frac{1}{3}} \cdot (8)^{-\frac{1}{3}}\)
- \(27^{\frac{1}{3}} = 3\) (since \(27 = 3^3\))
- \((8)^{-\frac{1}{3}} = \frac{1}{2}\) (since \(8 = 2^3\))

So:
\[
3 \cdot \frac{1}{2} = \frac{3}{2}
\]

### 4. Simplify the denominator \(\sqrt{\frac{4}{9}} - 4^0\)
- \(\sqrt{\frac{4}{9}} = \frac{2}{3}\)
- \(4^0 = 1\)

Thus:
\[
\frac{2}{3} - 1 = \frac{2}{3} - \frac{3}{3} = -\frac{1}{3}
\]

### 5. Combine everything:
The full expression is:
\[
3 + \frac{4}{5} + \frac{3}{2} \div \left(-\frac{1}{3}\right)
\]

#### 5.1 Simplify the division part:
\[
\frac{3}{2} \div \left(-\frac{1}{3}\right) = \frac{3}{2} \times \left(-3\right) = -\frac{9}{2}
\]

#### 5.2 Combine all the terms:
Now we have:
\[
3 + \frac{4}{5} - \frac{9}{2}
\]

To combine these, we need a common denominator. The least common denominator of 5 and 2 is 10:
- \(3 = \frac{30}{10}\)
- \(\frac{4}{5} = \frac{8}{10}\)
- \(-\frac{9}{2} = -\frac{45}{10}\)

So:
\[
\frac{30}{10} + \frac{8}{10} - \frac{45}{10} = \frac{30 + 8 - 45}{10} = \frac{-7}{10}
\]

### Final Answer:
The simplified value of the given expression is \(\frac{-7}{10}\).

---

Do you want more details on any part of the solution, or have any questions?

Here are five related questions:
1. How do negative exponents affect a fraction?
2. What is the cube root of a fraction like \(\frac{1}{64}\)?
3. How does division of fractions work?
4. Why is \(4^0 = 1\)?
5. How can we simplify expressions with mixed operations (multiplication, division, etc.)?

**Tip:** When working with expressions involving roots and exponents, always simplify the exponents first before handling other operations.

(1/9)^(-1/2) + 4^2 * ∛(1/64) * (25)^(-1/2) + 27^(1/3) * (8)^(-1/3) ÷ √(4/9) - 4^0

Step-by-step simplification of a complex algebraic expression involving exponents and radicals. The problem demonstrates the application of properties of exponents, including negative and fractional exponents, and is suitable for high school students in Grades 9-12.

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