Powers with Negative Exponents
In this section, we address the concept of negative exponents and how they relate to fractional representations. Understanding negative exponents is essential in mathematics as they enable us to express very small numbers concisely. Negative exponents indicate the reciprocal of the base raised to the respective positive exponent. For instance, the expression a^{-m} denotes 1/a^{m}. In examples from powers of 10, we observe that as an exponent decreases, the value becomes one-tenth of the value observed previously. This section provides the properties governing negative exponents alongside practical exercises to reinforce understanding.
Additionally, we explore the laws governing exponents that still hold true for negative values, such as the product rule, quotient rule, and power rule, establishing a uniformity in the application of these laws regardless of the exponent's value. Utilizing these laws, we illustrate complex exponent operations, express numbers in expanded form, and apply the concept of multiplicative inverses, presenting the relevance of exponents within larger mathematical contexts.