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Substitution in expressions is a fundamental concept in algebra that involves replacing variables with their given values. This technique allows us to evaluate expressions and solve equations by using known quantities instead of variables:

• First, identify the variables that you need to substitute. In the given exercise, we deal with the variable \(x\).
• Check the values you have for these variables. Here, \(x\) is given as 3.
• Take the original expression, \(x^{-4}\), and substitute the value of \(x\), giving us \(3^{-4}\).

By substituting, we transform the abstract expression into something numerical and directly solvable. It’s like translating a word problem into math language it makes further calculations possible!

Exponentiation is an essential operation in mathematics where a number, known as the base, is raised to a power, called the exponent. In simple terms:

• The base tells you what number you're multiplying.
• The exponent tells you how many times to multiply the base by itself.

For the expression \(3^{-4}\), we apply the exponentiation rule. However, since the exponent is negative, we need to use a specific rule: the negative exponent rule. This rule indicates that a base raised to a negative exponent is the reciprocal of the base raised to the positive of that exponent. Mathematically, \(a^{-n} = \frac{1}{a^n}\).
So, our task is to rewrite \(3^{-4}\) by converting the negative exponent into a positive one, leading to \(\frac{1}{3^4}\). This step transforms a potentially complex-looking expression into something easier to evaluate.

Mathematical operations like exponentiation involve performing detailed step-by-step calculations to arrive at a solution:

• After substituting and reformulating the expression into \(\frac{1}{3^4}\), calculate the positive exponent first.
• This means multiplying 3 by itself four times: \(3 \times 3 \times 3 \times 3 = 81\).

Next, fulfill the division in the reformulated expression. We do this by realizing that \(\frac{1}{3^4} = \frac{1}{81}\).
These operations show how breaking down a problem into a series of simpler steps can offer clarity. Each calculation plays a crucial part in solving the full problem, whether dealing with complex numbers or practicing basic arithmetic.