Final answer:
Negative exponents represent the reciprocal of the base raised to the positive exponent, indicating a division rather than multiplication. The rules for arithmetic operations, like adding and subtracting, apply to negative exponents in the same way as to positive exponents.
Step-by-step explanation:
Negative exponents are closely related to their whole number counterparts in that they represent the reciprocal of the base raised to the positive exponent. In essence, a negative exponent indicates a division by that base raised to the positive exponent. For instance, if we consider x-n, this is equivalent to 1 / xn, where 'n' is a positive integer. This relationship means that as the exponent decreases in magnitude (goes from positive to negative), the result moves from a multiplication to a division.
Adding exponents follows the same arithmetic rules regardless of whether the exponents are positive or negative. When we divide numbers with the same base, we subtract the exponents, and this includes negative exponents. If dealing with scientific notation, negative exponents indicate a small decimal number. For example, 106 / 103 is 106-3 or 103, demonstrating the subtraction of exponents during division.
To understand this concept better, consider the scientific notation for small numbers using negative exponents, which is often used when dealing with very small quantities. The table of powers of 10 with negative exponents can help visualize this concept. Each negative exponent corresponds to a decimal number less than one, indicating a division by a certain power of ten.