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Square roots are a fundamental concept in algebra that involve finding a number which, when multiplied by itself, results in the original number. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9. Square roots are often represented with the radical symbol \(\sqrt{}\).
In expressions involving square roots, it's crucial to understand how to simplify and manage them, especially when dealing with algebraic variables. For instance, when working with \(\sqrt{x^2}\), the result simplifies to \(x\), assuming \(x\) represents a positive value. This simplification occurs because \(x \times x = x^2\).
Another key point is simplifying expressions like \(\sqrt{9x^2}\). Here, you can separate the square root into components: \(\sqrt{9} \cdot \sqrt{x^2}\), which simplifies to 3 and \(x\), respectively. Always remember the rule: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).
This knowledge is essential when working through problems that involve the multiplication or distribution of radicals.

The distributive property is a useful algebraic rule that simplifies expressions and equations by distributing one term across terms inside parentheses. It can be expressed as \(a(b + c) = ab + ac\).
In the given problem, \(2 \sqrt{x}(\sqrt{9x} + 3)\), the distributive property allows us to "spread" \(2 \sqrt{x}\) to each term within the parentheses:

• Multiply \(2 \sqrt{x}\) by \(\sqrt{9x}\)
• And, \(2 \sqrt{x}\) by 3

This process results in: \(2\sqrt{x} \cdot \sqrt{9x} + 2\sqrt{x} \cdot 3\).
The distributive property is particularly powerful when dealing with variables and complex expressions, as it breaks down the terms into manageable parts, allowing for further simplification and solution.

The multiplication of radicals, or square roots, simplifies the interaction of numbers or expressions under square root symbols. When multiplying radicals, you use the rule: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). This approach helps simplify expressions that include radicals.
In our exercise, when we multiply \(\sqrt{x}\) by \(\sqrt{9x}\), we combine the terms under a single square root: \(\sqrt{x \cdot 9x}\), which becomes \(\sqrt{9x^2}\). This simplification is crucial in solving or simplifying algebraic expressions.
Another example from the exercise is the simplification of \(2 \cdot \sqrt{9x^2}\). By simplifying \(\sqrt{9x^2}\) to \(3x\), due to both the square root of 9 resulting in 3 and \(x^2\) simplifying to \(x\), the expression becomes \(6x\) when multiplied by 2.
This knowledge aids in maneuvering through complex expressions, breaking them down methodically to reach a final simplified form.