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Understanding how to manage exponents in fractions is crucial for simplifying complex expressions. When raising a fraction to a power, like \( \left(\frac{3}{5}\right)^{2} \), it means we multiply the fraction by itself that many times. In this case, it's \( \left(\frac{3}{5}\right) \times \left(\frac{3}{5}\right) \).

Each part of the fraction—the numerator (top number) and the denominator (bottom number)—is raised to the power independently. Therefore, \( (\frac{3}{5})^{2} \) becomes \( \frac{3^{2}}{5^{2}} \), which simplifies to \( \frac{9}{25} \).

Multiplication is the key here, and keeping the exponent rules straight when dealing with fractions helps prevent mistakes. It's also important to simplify the fraction if possible after exponentiation, but in this instance, \( \frac{9}{25} \) is already in its simplest form.

When working with fractions, especially in subtraction and addition, it is essential to have common denominators. The denominator is the 'common ground' that allows us to compare or combine fractions. If the denominators are different, we can't simply subtract the numerators.

In our exercise, we have the fractions \( \frac{9}{25} \) and \( \frac{3}{10} \). To subtract these, we need to make their denominators the same. The Lowest Common Denominator (LCD) for 25 and 10 is 25, so we multiply both the numerator and denominator of the second fraction, \( \frac{3}{10} \), by 2.5 (or equivalently multiply by 5 to keep it in whole numbers). As a result, \( \frac{3}{10} \) becomes \( \frac{3 \times 5}{10 \times 5} \), or \( \frac{15}{25} \). We've now created equivalent fractions with a common denominator, which makes subtraction possible.

Subtracting fractions with a common denominator is straightforward. Once you have the same denominator for both fractions, as we achieved in our example with \( \frac{9}{25} \) and \( \frac{15}{25} \), all you need to do is subtract the numerators.

The resulting fraction after this operation will have the same denominator as the original fractions. Therefore, \( \frac{9}{25} - \frac{15}{25} \) gives us \( \frac{9 - 15}{25} \) which simplifies to \( \frac{-6}{25} \). Notice that the denominator remains unchanged.

It's essential to simplify the result if possible. However, in the final answer \( \frac{-6}{25} \) none of the numbers share a common factor other than 1, so this fraction is indeed in its simplest form.