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Exponentiation is the mathematical operation involving the raising of a number to a power. The number being raised is called the base, and the power it is raised to is called the exponent. For example, in \((-z)^5\), \(-z\) is the base, and 5 is the exponent. When you raise a base to an exponent, you multiply the base by itself as many times as the value of the exponent. For instance:

• \((-z)^2 = (-z) \times (-z)\)
• \((-z)^3 = (-z) \times (-z) \times (-z)\)
• \((-z)^5 = (-z) \times (-z) \times (-z) \times (-z) \times (-z)\)

In our exercise, \((-z)^5\) simplifies as follows: \(-1 \times z \times z \times z \times z \times z = -z^5\). Here, -1 is multiplied by \(z^5\), resulting in \(-z^5\).

Rational expressions are fractions that consist of a numerator and denominator, both of which are polynomials. In simpler terms, they consist of ratios of algebraic expressions. For example, consider the rational expression \( \frac{x^2 + 3x - 4}{x - 1} \). Here,

• The numerator is \( x^2 + 3x - 4\)
• The denominator is \( x - 1 \)

These expressions can be simplified by factoring the polynomials and canceling out common factors. Keep in mind that the denominator of a rational expression cannot be zero, as division by zero is undefined. In our exercise, we analyze \( \frac{-z^5}{-z^5} \). Since both the numerator and the denominator are the same non-zero value, the rational expression simplifies to 1.

Simplifying algebraic fractions involves reducing them to their simplest forms. This process may involve factoring polynomials, canceling common factors, or simply performing basic arithmetic operations. Let's break down the steps: 1. **Factor** the numerator and the denominator to identify any common factors.
2. **Cancel** out any common factors present in both the numerator and the denominator.
For example, in the fraction \( \frac{(x+2)(x-2)}{x+2} \), the common factor \((x+2) \) in both the numerator and denominator can be canceled to simplify the fraction to \( x-2 \). In the provided exercise, the initial fraction is \( \frac{(-z)^5}{-z^5} \). Simplifying the numerator and recognizing the equivalency in the denominator allows us to cancel out terms, leading directly to: \( \frac{-z^5}{-z^5} = 1 \). Thus, the expression is simplified to its simplest form.