The expression in the image is:

$\left( \left( \frac{3}{5} \right)^0 + \left( \frac{3}{5} \right)^1 + \left( \frac{3}{5} \right)^2 \right) \div \left( \frac{7}{5} \right)^2$

Let's break it down step by step:

Step 1: Simplify the terms inside the brackets.

1. $\left( \frac{3}{5} \right)^0 = 1$ (any number raised to the power of 0 is 1).
2. $\left( \frac{3}{5} \right)^1 = \frac{3}{5}$.
3. $\left( \frac{3}{5} \right)^2 = \frac{9}{25}$ (square the numerator and denominator).

Now, summing these: $1 + \frac{3}{5} + \frac{9}{25}$

To add these, we need a common denominator. The least common denominator (LCD) is 25, so: $1 = \frac{25}{25}, \quad \frac{3}{5} = \frac{15}{25}$

Thus, the sum becomes: $\frac{25}{25} + \frac{15}{25} + \frac{9}{25} = \frac{49}{25}$

Step 2: Simplify the denominator.

The denominator is $\left( \frac{7}{5} \right)^2 = \frac{49}{25}$.

Step 3: Perform the division.

We now have: $\frac{49}{25} \div \frac{49}{25}$

Dividing two equal fractions gives 1: $\frac{49}{25} \div \frac{49}{25} = 1$

Thus, the simplified value of the given expression is 1.

Would you like any additional details or have questions about the process?

Here are 5 related questions to explore further:

1. What is the general rule for adding fractions with different denominators?
2. How do you handle expressions with powers in both the numerator and denominator?
3. How can you apply this technique to more complex fractions?
4. What happens when you divide fractions with different values?
5. Can you explain why any non-zero number raised to the power of 0 equals 1?

Tip: When dealing with fractions and powers, always simplify both the numerator and denominator separately before combining them!