Let's break down the expression in the image you uploaded:

[ \left[ -3^0 \cdot \sqrt[4]{0.1^{\frac{1}{2}}} + (-3)^0 \cdot 64^{\frac{2}{3}} \right] \div \left( -\frac{1}{32} \right)^{-\frac{1}{5}} ]

Step-by-step analysis:

1. Simplify the first term:

   • $3^0 = 1$, so the first term becomes: $-1 \cdot \sqrt[4]{0.1^{\frac{1}{2}}}$
   • Now, simplify $0.1^{\frac{1}{2}}$, which is the square root of 0.1: $\sqrt{0.1} \approx 0.3162$
   • Take the fourth root of this result: $\sqrt[4]{0.3162} \approx 0.796$
   • Therefore, the first term becomes: $-0.796$

2. Simplify the second term:

   • $(-3)^0 = 1$, so this term becomes: $1 \cdot 64^{\frac{2}{3}}$
   • Simplify $64^{\frac{2}{3}}$. This means taking the cube root of 64, then squaring it: $\sqrt[3]{64} = 4$ Then square this: $4^2 = 16$
   • Therefore, the second term is: $16$

3. Combine both terms: $-0.796 + 16 = 15.204$

4. Simplify the denominator:

   • Simplify $\left( -\frac{1}{32} \right)^{-\frac{1}{5}}$. First, rewrite the negative exponent: $\left( -\frac{1}{32} \right)^{-\frac{1}{5}} = \left( -32 \right)^{\frac{1}{5}}$
   • $32 = 2^5$, so: $\left( 2^5 \right)^{\frac{1}{5}} = 2$
   • Therefore, the denominator is: $2$

5. Final calculation: $\frac{15.204}{2} = 7.602$

Thus, the value of the expression is approximately 7.602.

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

Here are some related questions you might find interesting:

1. How do negative exponents work in mathematical expressions?
2. What is the general method to simplify expressions involving roots and powers?
3. How do we handle fractional exponents in calculations?
4. What does the concept of cube roots and fourth roots mean in terms of powers?
5. How can we evaluate expressions with both exponents and roots efficiently?

Tip: When dealing with roots and fractional exponents, always remember that fractional exponents can be broken down into root and power components, making them easier to simplify.