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Understanding algebraic operations is essential for simplifying expressions like the given square roots. When we come across square roots, such as \( \sqrt{32} \) or \( \sqrt{18} \) in the exercise, we use algebraic operations to break them down into simpler forms. This often involves identifying perfect square factors. For instance, 32 can be broken down into \( 16 \times 2 \) and since \( 16 \) is a perfect square (as \( 4^2 = 16 \) ), we can take it out of the root. Thus, \( \sqrt{32} \) simplifies to \( 4\sqrt{2} \). Similarly, \( \sqrt{18} \) simplifies to \( 3\sqrt{2} \) as it can be written as \( 9 \times 2 \) and 9 is a perfect square (\( 3^2 = 9 \) ). Simplifying square roots simplifies the forthcoming addition process.

When tasked with adding fractions, the crux of the process is often finding common denominators. This allows us to aggregate the fractions and perform straightforward addition of their numerators. A common denominator is typically the least common multiple of the existing denominators—in this case, 5 and 7.

To find this for 5 and 7, since they are prime relative to each other (they share no common factors apart from 1), we simply multiply them to get 35. Following this, we adjust the fractions by multiplying the numerators and the denominators in such a way that we do not change the value of the fractions but achieve equivalent fractions with a common denominator. Hence, \( \frac{4\sqrt{2}}{5} \) becomes \( \frac{28\sqrt{2}}{35} \) and \( \frac{3\sqrt{2}}{7} \) becomes \( \frac{15\sqrt{2}}{35} \).

Adding fractions requires the denominators of the fractions to be the same. Once we have the same denominators, as achieved through the common denominator process, it's a matter of adding the numerators. Keep in mind that we can only add fractions directly when they contain like terms.

In our exercise, both numerators include \( \sqrt{2} \) which allows us to add the coefficients in front of the radical sign, here being 28 and 15. This summation gives us 43, leading to a combined fraction of \( \frac{43\sqrt{2}}{35} \). A good habit is to check if it's possible to simplify further; in this case, since 43 and 35 share no common factors besides 1, the fraction is already in its simplest form.