Final answer:
When adding or subtracting scientific notation, the exponents must be the same because we combine quantities of the same order of magnitude. For multiplication and division, different exponents can be used since these operations combine magnitudes and effectively sum or subtract the exponents.
Step-by-step explanation:
To understand why in adding and subtracting scientific notation, the exponents must be the same while in multiplying and dividing they can differ, we need to consider the nature of these operations with powers of ten. In addition and subtraction, we are combining or removing quantities that need to be of the same order of magnitude, hence the exponents must match. For example, you cannot directly add 2.3 x 10³ and 5.6 x 10´ without adjusting them to the same power of ten, usually by converting both numbers to the lower power. This is the arithmetic equivalent of making sure units match when combining measurements.
However, when we multiply or divide numbers in scientific notation, we are essentially combining or dividing their magnitudes and summing or subtracting their exponents, which means different powers of ten can be easily accommodated. The process is demonstrated by the equation (3 x 10µ) x (2 x 10°) = 6 x 10µ. Here, the coefficient (numbers out front) are multiplied producing 6, and the exponents are simply added together, resulting in a final exponent of 5.