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When tasked with multiplying square roots, it can initially seem daunting, but it's quite straightforward once you understand the concept. The rule to remember is this: when you multiply the square roots of two numbers, you are actually taking the square root of their product. This is especially useful in simplifying calculations that involve square roots.
For instance, if we consider the expression \( \sqrt{2} \times \sqrt{7} \), by applying this rule, it simplifies to \( \sqrt{2 \times 7} = \sqrt{14} \). The same applies to \( \sqrt{2} \times \sqrt{10} \), which becomes \( \sqrt{2 \times 10} = \sqrt{20} \).
This property allows us to combine tasks involving roots into a single, streamlined operation. Remember, this can only be used when multiplying—division, subtraction, and addition follow different rules.

After multiplying the square roots, you might end up with expressions that can be simplified further. This involves breaking down the square root into simpler parts. For example, \( \sqrt{20} \) can be broken down into \( \sqrt{4 \times 5} \).
Notice that \( \sqrt{4} \) is 2 neatly, since 4 is a perfect square. Thus, \( \sqrt{20} = 2\sqrt{5} \). This technique of simplifying is very helpful in reducing expressions to their simplest form, making calculations easier.
Always check if the number under the square root sign can be factored into a perfect square and another integer. This will enable you to simplify any complex expression into a more manageable form.

An algebraic expression like \( \sqrt{2}(\sqrt{7} + \sqrt{10}) \) involves a combination of numbers and algebraic terms. Here, you need to apply both arithmetic with numbers and properties of operations, such as the distributive property.
The distributive property helps by spreading out the operation to each term within the parentheses: \( \sqrt{2} \times (\sqrt{7} + \sqrt{10}) = \sqrt{2} \times \sqrt{7} + \sqrt{2} \times \sqrt{10} \).
Handling algebraic expressions often requires using multiple principles simultaneously to simplify or solve them effectively. Whether multiplying terms or simplifying results, it's essential to keep these foundational arithmetic and algebraic rules in mind to decode the problems easily.