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Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. It signifies repeated multiplication of the base. For instance, raising a number to the power of 2 means multiplying it by itself once. In our example, the expression \(16^{-\frac{1}{4}}\) involves raising 16 to a negative fractional power.
But what does a negative exponent mean? A negative exponent indicates that instead of multiplying the base multiple times, we divide 1 by the base raised to the corresponding positive exponent.

• A positive exponent means normal multiplication.
• A negative exponent means we take the reciprocal of the base raised to the positive exponent.

Understanding exponentiation is key as you will often see it in math in different contexts, like radical expressions and polynomials.

When we talk about the fourth root, we mean a number that, when multiplied by itself four times, equals the initial number. The fourth root of a number \(x\) is denoted as \(\sqrt[4]{x}\).
In the exercise, finding the fourth root of 16 means determining what number multiplied by itself four times equals 16. This number is 2, because:

• \(2 \times 2 \times 2 \times 2 = 16\)

Taking the fourth root is an inverse operation to raising a number to the fourth power. This is similar to how multiplication is the inverse of division. Understanding root calculations is crucial as it often appears in simplifying radical expressions and solving equations.

Simplifying radicals involves rewriting a radical expression in its simplest form. This is important for making expressions easier to work with. The key is to find the simplest base value or expression that, when raised to the required power, results in the original number under the radical.
In the given exercise:

• The expression \(\frac{1}{\sqrt[4]{16}}\) acknowledges \(\sqrt[4]{16} = 2\) as 2 is the simplest form.
• Thus, \(\frac{1}{\sqrt[4]{16}} = \frac{1}{2}\).

This process simplifies calculations and expressions, and understanding this concept helps in algebraic manipulations and solutions. Always ensure the simple radical form is accurate to avoid errors in your work.