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When it comes to multiplying square roots, it's important to start with understanding how they interact with each other. Consider the square roots \( \sqrt{a} \) and \( \sqrt{b} \). The rule for multiplying these is:

• \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \)

This rule is very helpful because it allows us to handle large expressions more comfortably.
In our exercise, we multiply \( \sqrt{5x} \) by \( \sqrt{11y} \). According to the rule, we combine the expressions under a single square root: \( \sqrt{(5x) \cdot (11y)} \).
This simplification aids in streamlining the expression for further manipulation, especially in algebraic contexts where cleaner expressions are often easier to work with.

Radical multiplication involves multiplying expressions under a radical, which could be square roots or other roots. In our exercise, we multiply the radicals \( \sqrt{5x} \) and \( \sqrt{11y} \).
The main principle here is similar to regular multiplication, but applied within the bounds of a square root. The radicals are combined under one radical sign as:

• \( \sqrt{5x} \cdot \sqrt{11y} = \sqrt{(5x)(11y)} \)

Understanding this process is crucial because it simplifies the work you need to do with expressions, preparing you for more complex algebraic tasks. Keep in mind this principle is broad and not restricted to just square roots, expanding to other types of roots too.
Remember, when two radicals are multiplied, their inner terms are multiplied together independently and placed back under a single radical.

Simplifying algebraic expressions is all about making them easier to understand and manipulate. In this context, after multiplying the square roots, the expression \( \sqrt{(5x)(11y)} \) can still be simplified.
Here’s a quick breakdown of the simplification process:

• Calculate the product of the numerical coefficients: \( 5 \cdot 11 = 55 \).
• Multiply the variables: \( x \cdot y \).
• Combine them under the square root to form \( \sqrt{55xy} \).

Simplification helps in various ways, like reducing complexity in equations and making them more manageable to solve. Always seek further simplification where possible to achieve the clearest form of an expression, just like in our example, where merging into \( \sqrt{55xy} \) is more concise than keeping separate radical terms.