The given mathematical expression in the image is:

$(0.04)^{-1.5} - (0.125)^{-\frac{2}{3}}$

Step 1: Evaluate each term separately

1. Evaluate $(0.04)^{-1.5}$:

   • Rewrite $0.04$ as $\frac{1}{25}$, so: $(0.04)^{-1.5} = \left(\frac{1}{25}\right)^{-1.5}$
   • Using the rule $a^{-n} = \frac{1}{a^n}$, this becomes: $\left(\frac{1}{25}\right)^{-1.5} = 25^{1.5}$
   • $25^{1.5} = \sqrt{25}^3 = 5^3 = 125$.

   So, $(0.04)^{-1.5} = 125$.

2. Evaluate $(0.125)^{-\frac{2}{3}}$:

   • Rewrite $0.125$ as $\frac{1}{8}$, so: $(0.125)^{-\frac{2}{3}} = \left(\frac{1}{8}\right)^{-\frac{2}{3}}$
   • Using the rule $a^{-n} = \frac{1}{a^n}$, this becomes: $\left(\frac{1}{8}\right)^{-\frac{2}{3}} = 8^{\frac{2}{3}}$
   • $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$.

   So, $(0.125)^{-\frac{2}{3}} = 4$.

Step 2: Combine results

Now substitute the values back into the original expression:

$(0.04)^{-1.5} - (0.125)^{-\frac{2}{3}} = 125 - 4 = 121$

Final Answer:

$\boxed{121}$

Would you like more details on the calculations or further examples?

Related Questions:

1. What are the steps to calculate fractional exponents like $a^{m/n}$?
2. How do negative exponents work with fractions?
3. Can you explain how to simplify complex fractional bases like $0.04$?
4. How can we rewrite numbers in different forms to simplify calculations?
5. What are common mistakes to avoid when dealing with fractional and negative exponents?

Tip:

Always rewrite numbers in fractional or exponential form to simplify calculations with exponents.