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When we talk about the power of a number, we're discussing how many times a number (called the base) is multiplied by itself. For example, in the expression \(7^2\), the base is 7 and it is raised to the power of 2. This means that 7 is multiplied by itself once to get 49. Calculating powers is a fundamental process in mathematics and helps simplify larger problems into smaller, manageable multiplications.
A negative sign in the exponent changes things a bit. Instead of multiplying, a negative exponent signifies division. For instance, if you have \(a^{-n}\), it translates to \(\frac{1}{a^n}\). Therefore, negative exponents, rather than increasing the size of the number, make it a fraction less than one.

Substitution in expressions is a key technique in algebra, where you replace a variable with a given number in order to evaluate the expression. This allows you to find the value of expressions for specific variable sets. For example, if we're given that \(s = -2\) and need to evaluate \(7^s\), we substitute \(-2\) in place of \(s\), transforming the expression into \(7^{-2}\).
This process is essential for solving equations and making sense of mathematical models. By using substitution, you convert abstract expressions into concrete values, making calculations and evaluations straightforward. It’s like connecting the numbers and the variables to know exactly what you are dealing with.

The rules of exponents are a set of guidelines that help simplify expressions involving powers. Using these rules correctly allows for quick and accurate evaluations and reductions. Here are the main rules you need to remember:

• **Product of Powers:** \(a^m \times a^n = a^{m+n}\).
• **Quotient of Powers:** \(\frac{a^m}{a^n} = a^{m-n}\).
• **Power of a Power:** \((a^m)^n = a^{m \times n}\).
• **Negative Exponent Rule:** \(a^{-n} = \frac{1}{a^n}\).

In our exercise, the Negative Exponent Rule is particularly useful. By rewriting \(7^{-2}\) as \(\frac{1}{7^2}\), this rule helps us turn what seems like a complex expression into a straightforward division. Understanding these rules lets you simplify complex expressions into recognizable patterns, making advanced calculations easier.