Final answer:
Simplifying expressions with negative exponents involves recognizing them as indicative of division or reciprocals (1/x^n), whereas positive exponents suggest multiplication. Simplification must result in an acceptable form, often using scientific notation, and arithmetic rules of adding and subtracting exponents apply during multiplication and division.
Step-by-step explanation:
Simplifying an expression with negative exponents differs from simplifying with positive exponents in that negative exponents indicate division or a reciprocal relationship, rather than multiplication. For example, when dealing with a negative exponent such as x-n, it implies 1/xn. Conversely, a positive exponent denotes repeated multiplication. When simplifying expressions with mixed exponents, we apply the regular arithmetic rules of addition and subtraction to the exponents themselves: we add the exponents when multiplying and subtract them when dividing bases. It is critical that we express the final result in an acceptable form, where numbers greater than 10 or less than 1 must be properly notated in scientific notation. This often involves adjusting the coefficient to a number between 1 and 10 and modifying the exponent accordingly.
To illustrate, dividing numbers with exponents requires subtracting the exponents of like bases. For example, 106 รท 103 becomes 10(6-3) or 103. Also, small numbers can be expressed in scientific notation with negative exponents to maintain proper form. It's important to remember that as the numeric value of the exponent decreases, the resulting value of the base decreases in magnitude, and vice versa.