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.