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Square root multiplication might seem tricky at first, but it's quite simple once you understand the basic concept. When you multiply two square roots together, there's a useful property you can rely on: the product of two square roots is the square root of their product. For example, if you have \(\text{sqrt}(a) \cdot \text{sqrt}(b)\), this is equal to \(\text{sqrt}(a \cdot b)\). This property makes things much easier because you can combine the two numbers under a single square root symbol. In the given exercise, we start with \(\text{sqrt}(12) \cdot \text{sqrt}(48)\). Applying the property, we can transform this into \(\text{sqrt}(12 \cdot 48)\). This step allows us to simplify our calculations.

The product property of square roots is what allows us to combine two square roots into one. This is a very powerful tool for simplification. Let’s break down why this works. When you multiply \(\text{sqrt}(a)\) and \(\text{sqrt}(b)\), what you’re really doing is multiplying the values inside the radicands first—those are the numbers inside the square root symbols. Therefore, \(\text{sqrt}(12) \cdot \text{sqrt}(48)\) becomes \(\text{sqrt}(12 \cdot 48)\), which results in \(\text{sqrt}(576)\). Therefore, by using this property, our complex expression becomes much simpler to handle.

Simplifying square roots is the final step and often the most satisfying. Once you have a single square root, like \(\text{sqrt}(576)\), you need to find the simplest form of the square root. This means we have to express \(\text{sqrt}(576)\) in its most reduced form. Recognize that 576 is a perfect square: \(\text{576} = 24^2\). Knowing the perfect squares is very useful here because it makes simplification straightforward. Since 576 can be written as \(\text{24}^2\), the square root of \(\text{576}\) is simply \(\text{24}\). Thus, \(\text{sqrt}(576) = 24\). Putting all steps together, we arrive at the final simplified form of the original multiplication of square roots. Each step helps to clarify and reduce the expression into a more manageable number.