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Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. In the context of the provided exercise, algebra involves finding the value of an unknown variable, represented by the letter 'a', through the use of algebraic expressions and equations.

The exercise presents a scenario where two numbers are given, and the geometric mean is provided. The geometric mean is an algebraic concept used to find the average ratio of a set of numbers. To solve for the variable 'a', we need to translate the word problem into an algebraic equation, which brings us to our next core concept, solving equations.

Solving equations is a fundamental part of algebra and involves finding the value or values of variables that make an equation true. To solve the equation for 'a' in the exercise, the first step is to recognize the geometric mean formula and set up the equation \(\sqrt{12 \times a} = 27\).

Next, we need to isolate the variable 'a', which is achieved by squaring both sides to remove the square root, resulting in the equation \(12 \times a = 27^2\). The squaring process is a common technique used when dealing with square roots in equations. Once we have a basic algebraic equation, we can use simple arithmetic operations, like division, to solve for 'a'.

Simplify and Calculate

The last steps involve simplifying the squared number and then dividing both sides by 12 to find the exact value of 'a'. By systematically approaching the equation, each operation moves you closer to the solution.

Square roots are mathematical operations that answer the question: 'What number, when multiplied by itself, gives the original number?'. Square roots are symbolized by the radical sign \(\sqrt{}\). In our exercise, the geometric mean is presented as \(\sqrt{12 \times a}\), which indicates that we need to find a number that, when squared, gives us the product of 12 and 'a'.

To solve for 'a', we must manipulate the equation by squaring both sides as shown in the second step of the solution. This process eliminates the square root and allows us to deal with a simpler algebraic equation. Understanding square roots is crucial since they frequently appear in various areas of mathematics, including geometry and algebra. The square root operation is the inverse of squaring a number, and it's important to remember that only non-negative numbers have real-numbered square roots.