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Exponents are a fundamental concept in mathematics. An exponent indicates how many times a number (the base) is multiplied by itself. For example, in the expression \(2^3\), 2 is the base and 3 is the exponent, meaning \(2 \times 2 \times 2 = 8\). Exponents greatly simplify expressing large numbers or repeated multiplication. When seen in general terms: \(a^n\) means you multiply \(a\) by itself \(n\) times. Special cases include \(a^1 = a\) and \(a^0 = 1\). This simple principle is the foundation for understanding more complex uses, such as negative exponents and algebraic substitution.

Negative exponents represent the reciprocal of a number raised to the corresponding positive exponent. For example, \(a^{-n} = \frac{1}{a^n}\). This is especially useful in transforming division problems into multiplication problems. Consider the example from the exercise: \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\). Likewise, \((-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} = -\frac{1}{8}\). Remember that when dealing with negatives, the placement of the negative sign is crucial. If \(a\) is a negative base, make sure the entire base is in parentheses.

Substitution in algebra involves replacing a variable with a given number to simplify or solve an expression. Here, we are replacing \(x\) with given values and simplifying accordingly. For example, to evaluate \(4x^{-3}\) for \(x=2\), we substitute 2 in place of \(x\): \(4(2^{-3})\). We already know \(2^{-3} = \frac{1}{8}\), so the expression becomes \(4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2}\). Doing the same for \(x=-2\), \(4((-2)^{-3}) = 4 \times -\frac{1}{8} = -\frac{4}{8} = -\frac{1}{2}\). Practice this concept using different numbers to better understand how substitution works in various contexts. Always pay attention to the signs and values involved in each substitution for accurate results.