Final answer:
The value of 'a' in scientific notation is constrained between 1 and 10 to maintain clear, standardized, and unique representation of numbers. The coefficient 'a' works with the exponent 'b' in the scientific notation formula a x 10^b to simplify calculation and manipulation of large and small numbers.
Step-by-step explanation:
The reason that a needs to be between 1 and 10 in scientific notation is to ensure that every number is expressed in a consistently clear and standardized format. In scientific notation, represented as a x 10^b, a is referred to as the coefficient and it always takes a value greater than or equal to 1 and less than 10. This rule prevents ambiguity and maintains the uniqueness of the scientific notation for any given number.
For example, the number 1,372,568 would be written in scientific notation as 1.372568 x 10^6. By moving the decimal point six places to the left, the coefficient (1.372568) falls within the required range, and it is then multiplied by 10 to the sixth power to restore the original number's magnitude.
When multiplying and manipulating numbers in scientific notation, this standard keeps calculations straightforward. Instead of working with unwieldy long numbers, only the coefficients are multiplied and the exponents are added, as seen in the example (3 x 10^5) x (2 x 10^2) = 6 x 10^7.