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Radical expressions are mathematical expressions that include a radical symbol, which indicates the root of a number. The most common radical expression is the square root, denoted as \(\sqrt{x}\), where \(x\) is any nonnegative real number. For instance, in the expression \(\[\sqrt{51+13}\]\), we are dealing with a square root radical.
When simplifying radical expressions, it's essential to remember that the operation under the radical sign (inside the square root) should be performed first. That’s because the square root symbol encompasses everything within, and we must simplify that value before further interpreting the radical. In our example, the expression under the radical is simplified by performing the addition of 51 and 13 to yield 64, which can then be further simplified as radical expressions often involve perfect squares that have whole number square roots.

Square root calculations involve finding a value that, when multiplied by itself, gives the original number under the square root sign. For the expression \(\[\sqrt{51+13}\]\), once we have added the numbers to get 64, we need to find the square root of 64.

A square root can be determined either through estimation, prime factorization, or by using a calculator. In our case, recognizing that 64 is a perfect square is crucial for quick calculation, as it's equal to \(8^2\). Therefore, \(\sqrt{64} = 8\). It is helpful to remember the square roots of common perfect squares, such as 1, 4, 9, 16, 25, 36, 49, and so on, to make such calculations easier and faster.

Real numbers encompass all the numbers on the number line, including both rational numbers (which can be expressed as fractions, such as 1/2, and integers like -5, 0, 2) and irrational numbers (which cannot be precisely expressed as fractions and whose decimal forms are non-repeating and non-terminating, such as \(\pi\) and the square root of non-perfect squares).

In the context of square roots, every positive real number has a positive square root, known as the principal square root. For example, the square roots of 64 are 8 and -8, but the principal square root is 8, which is the positive value. It’s important to note that the square root of a negative number is not a real number; it's an imaginary number. In our exercise, because the expression under the square root was a positive number (64), the square root is a real number. This approach helps you to affirm if the result of a square root calculation will result in a real number.