Compute \(\left([3]_{73}\right)^{12}\) by raising 3 to the \(12^{\text {th }}\) power.

The modulus operation, often referred to as 'mod operation', plays a crucial role in various mathematical and computational fields. At its core, it's a division operation that, instead of providing the quotient, gives us the remainder after division. For instance, if you were to divide 10 by 3, the quotient would be 3, but the modulus operation would give you the remainder, which is 1. This is often denoted as 10 mod 3 = 1.

It's widely used in programming for tasks such as ensuring numbers fall within a certain range (like looping back to the start of a list after reaching the end) and in cryptography for securing information. The simplicity of the modulus operation is deceptive; although the concept is accessible to beginners, its application can become quite complex. Especially, it is indispensable in modular arithmetic, where the principle of 'wrap around' is employed, akin to the way a clock resets after hitting 12.

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. The exponent denotes how many times the base is multiplied by itself. For example, when we say 3 to the power of 4, expressed as \(3^4\), this means 3 multiplied by itself 4 times: 3 × 3 × 3 × 3, which equals 81.

Exponentiation is fundamental in various disciplines such as physics, engineering, and computer science because it describes exponential growth or decay, compound interest, and the behavior of waves, among other phenomena. With larger exponents, the number grows rapidly and can be used to represent very large quantities in a compact form. But when combined with the modulus operation, as in modular exponentiation, it allows operations on large numbers to be manageable and remain within practical computational limits.

Remainder computation is the process of finding the remainder of a division operation. When dividing a larger number by a smaller number, unless the larger is a multiple of the smaller, there will typically be some amount left over – this is the remainder. In mathematical terms, when we divide an integer a by an integer b, the remainder is the difference between a and the largest multiple of b that is less than or equal to a.

For example, with 20 ÷ 7, the largest multiple of 7 that does not exceed 20 is 14 (since 7 × 2 = 14), and so the remainder is 6, because 20 - 14 = 6. This calculation is critical in modular arithmetic, where the focus is not on the quotient but on the remainder. This process becomes especially important in computer science for designing algorithms that deal with large numbers in encryption, random number generation, and hashing, where dealing with remainders is often more relevant than dealing with quotients.