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Understanding the distributive property is vital for simplifying algebraic expressions, especially when dealing with radical expressions. The distributive property states that for any numbers a, b, and c, the equation a(b + c) equals ab + ac. In other words, when a number outside of the parenthesis is multiplied by a sum inside the parenthesis, you must multiply it by each term within the parenthesis separately.

For the example \(\sqrt{7}(9+\sqrt{7})\), we apply the distributive property by multiplying \(\sqrt{7}\) with both 9 and \(\sqrt{7}\), leading to two terms: \(9\sqrt{7}\) and \(\sqrt{7}\cdot\sqrt{7}\). This property simplifies the process of dealing with radical expressions and ensures that every term is addressed correctly.

Square roots are fundamental in algebra and they represent the opposite operation of squaring a number. Essentially, the square root of a number x, denoted as \(\sqrt{x}\), is the value that, when multiplied by itself, gives the original number x. For instance, if we have \(\sqrt{9}=3\) because 3 multiplied by 3 equals 9.

In the example above, we see \(\sqrt{7}\cdot\sqrt{7}\), which simplifies to 7 because \(\sqrt{7}\) is the value that, when squared, equals 7. It's important to become comfortable with this concept as it frequently appears in algebraic expressions involving radicals. Recognizing when a radical can be simplified in this way can make solving these expressions significantly easier.

Algebraic expressions are combinations of variables, numbers, and operations that represent a particular concept or quantity. In the context of the given exercise, we have an algebraic expression that involves both numerical and radical components. Simplifying such expressions is a key skill in algebra, which often involves combining like terms and using properties of arithmetic correctly.

When simplifying the expression \(9\sqrt{7} + 7\), notice that you cannot combine these terms further because they are not 'like terms.' A like term must have the exact same variable part, which here is differentiated by the presence of a radical. Recognizing and handling different terms correctly is crucial to ensure accurate simplification of algebraic expressions.