The expression in the image is:

$\left( \frac{5}{3} \right)^5 \cdot \left( \frac{5}{3} \right)^{2^2} : \left( 1 + \frac{2}{3} \right)^{2^{10}} - 1$

Let’s break it down step-by-step.

1. Simplify each exponent term:
   • $2^2 = 4$
   • $2^{10} = 1024$

So, the expression becomes:

$\left( \frac{5}{3} \right)^5 \cdot \left( \frac{5}{3} \right)^4 : \left( 1 + \frac{2}{3} \right)^{1024} - 1$

2. Combine the powers of $\frac{5}{3}$: Since we are multiplying powers of the same base, we can add the exponents:

   $\left( \frac{5}{3} \right)^{5 + 4} = \left( \frac{5}{3} \right)^9$

So now the expression is:

$\left( \frac{5}{3} \right)^9 : \left( 1 + \frac{2}{3} \right)^{1024} - 1$

3. Interpret the division and further simplify: Dividing by an exponent term is the same as subtracting the exponents when the bases are the same. However, since the bases are different, this step would generally involve keeping the expression as a fraction unless further context is provided.

4. Subtracting 1: After evaluating the division, the result has $- 1$ at the end, which will need to be subtracted from the final simplified fraction or result.

Would you like a detailed numeric approximation for the answer, or further simplification of the terms?

Here are five related questions to expand your understanding:

1. How do exponent rules apply to multiplication and division of terms?
2. What are properties of exponents in expressions with large powers, like $2^{10}$?
3. How can fractional bases be simplified in exponential form?
4. How does one evaluate large powers in exponential terms?
5. When should subtraction be applied after exponentiation and division in expressions?

Tip: Always apply exponent rules carefully to avoid misinterpreting multiplication and division, especially when the terms have the same base.