The mathematical expression in the image appears to be:

$(0.2)^{-2} + (5/3)^{-4}$

Let’s solve this step by step:

Step 1: Simplify $(0.2)^{-2}$

The negative exponent means we take the reciprocal and square it: $(0.2)^{-2} = \frac{1}{(0.2)^2} = \frac{1}{0.04} = 25$

Step 2: Simplify $(5/3)^{-4}$

Again, the negative exponent means we take the reciprocal and raise it to the positive power: $(5/3)^{-4} = \left(\frac{3}{5}\right)^4 = \frac{3^4}{5^4} = \frac{81}{625}$

Step 3: Add the two results

Now, sum the two terms: $(0.2)^{-2} + (5/3)^{-4} = 25 + \frac{81}{625}$

To add, convert $25$ to a fraction with the same denominator: $25 = \frac{25 \cdot 625}{625} = \frac{15625}{625}$

Now, add the fractions: $\frac{15625}{625} + \frac{81}{625} = \frac{15625 + 81}{625} = \frac{15706}{625}$

Step 4: Simplify the fraction (if possible)

The fraction $\frac{15706}{625}$ cannot be simplified further because 15706 and 625 do not have any common factors other than 1.

Final Answer:

$\frac{15706}{625} \quad \text{or approximately } 25.1296$

Would you like more explanation or help with related concepts? Here are some questions to explore:

1. How do negative exponents affect fractions?
2. What is the process of converting decimals to fractions for calculations like $(0.2)^{-2}$?
3. How can we determine if a fraction is in its simplest form?
4. What are the steps to convert a fraction result into a decimal?
5. How do powers of reciprocals work when simplifying expressions?

Tip: Always handle negative exponents by first flipping the base (reciprocal) before applying the power.