The following problems involve addition, subtraction, and...

The following problems involve addition, subtraction, and multiplication of radical expressions, as well as rationalizing the denominator. Perform the operations and simplify, if possible. All variables represent positive real numbers.
a. $(3 \sqrt{a})^{2}$
b. $(3+\sqrt{a})^{2}$

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Key Concepts

Radical Expressions

Radical expressions involve roots, most commonly square roots, and include terms under a radical symbol. In working with radical expressions, it is essential to understand how to manipulate and simplify them, particularly using properties like ?(a^2) = a for positive a and the fact that multiplication of radicals follows the rule ?a · ?b = ?(a*b). This concept forms the foundation for the operations in the given problem.

Exponentiation of Radical Terms

When raising an expression containing a radical to a power, the exponent is applied to both the coefficient and the radical. For example, in the expression (3?a)^2, the exponent 2 applies to both 3 and ?a separately, yielding (3^2)·((?a)^2). This technique leverages the properties of exponents to simplify expressions involving radicals.

Binomial Expansion Involving Radicals

Expanding a binomial that includes a radical, such as (3 + ?a)^2, uses the algebraic identity (x + y)^2 = x^2 + 2xy + y^2. This formula assists in systematically expanding and then simplifying the expression. Understanding this concept is important for correctly combining like terms and simplifying results when radicals are involved.

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