The image shows a complex expression involving fractions, powers, and parentheses. Let's rewrite the expression clearly before attempting to simplify or evaluate it.

The expression appears to be:

$\left( \frac{5}{3} \right)^5 \cdot \left( \frac{5}{3} \right)^2 : \left(1 + \frac{2}{3}\right)^6 - 1$

Let's break down and simplify each part:

Step 1: Simplify $\left( \frac{5}{3} \right)^5 \cdot \left( \frac{5}{3} \right)^2$

Since these two terms have the same base, we can add the exponents: $\left( \frac{5}{3} \right)^5 \cdot \left( \frac{5}{3} \right)^2 = \left( \frac{5}{3} \right)^{5+2} = \left( \frac{5}{3} \right)^7$

Step 2: Simplify $\left(1 + \frac{2}{3}\right)^6$

Inside the parentheses, add $1$ and $\frac{2}{3}$: $1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3}$ Thus, $\left(1 + \frac{2}{3}\right)^6 = \left( \frac{5}{3} \right)^6$

Step 3: Divide $\left( \frac{5}{3} \right)^7$ by $\left( \frac{5}{3} \right)^6$

Since both terms have the same base, we subtract the exponents: $\frac{\left( \frac{5}{3} \right)^7}{\left( \frac{5}{3} \right)^6} = \left( \frac{5}{3} \right)^{7-6} = \left( \frac{5}{3} \right)^1 = \frac{5}{3}$

Step 4: Subtract 1

Now we have: $\frac{5}{3} - 1 = \frac{5}{3} - \frac{3}{3} = \frac{2}{3}$

Final Answer

The simplified result of the expression is: $\frac{2}{3}$

Would you like further clarification or have any questions on this solution?

Here are five related questions you might explore:

1. How do exponent rules apply to terms with the same base?
2. What are the steps to simplify an expression involving powers and division?
3. How can fractions be simplified through addition or subtraction?
4. What are other methods to verify simplifications?
5. How can we extend these concepts to more complex expressions?

Tip: Always simplify expressions within parentheses first before handling exponents and other operations.