Scientific notation consists of two parts (1) the mantissa and (2) ten raised to a power ( power of ten for short). In a general sense, scientific notation is a special case of power of ten notation where the mantissa is restricted to one digit to the left of the decimal. Engineering notation is another special case where the power of ten exponent is zero or a multiple of three. Manipulating power of ten notation is an important skill for scientists and engineers. I have observed that many students coming from high school have been taught to do this by memorizing rules that involve counting zeros. While this is OK to a point, I believe there is a better way. I call it the "Order of Magnitude Method". Not only is it more general, it is based on simple logic rather than little rule memorizing.
First, we need to review some terminology. The phrase "order of magnitude" refers to a power of ten exponent change of +1 or -1. Making a number larger by an order of magnitude means adding +1 to the exponent. Making a number smaller by an order of magnitude means adding -1 to the exponent. Extending this idea a bit, to increase a number by three orders of magnitude means add three to the exponent. We will use this idea to make adjustments to numbers in power of ten notation.
Consider an example. Convert 34,167 to scientific notation. Here are the steps:
Write the number in power of ten notation by multiplying by 100. We have 34,167.100.
Mark the position where the decimal should be when finished. In this case, we want scientific notation so we want one digit to the left of the decimal. We have 3^4,167.100.
Note that moving the decimal point to the marked position makes the
mantissa smaller by four orders of magnitude. To compensate, we
must add +4 to the exponent of 10. We have
3.4167.100+4 = 3.4167.104
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Here's another example with a fractional value. Convert 0.0523 to scientific notation. It follows the same logical steps except that the mantissa is made larger by two orders of magnitude while adding -2 to the exponent.
We can easily reverse the process to change from scientific notation to standard notation. Change 4.823.102 to standard notation.
To zero the exponent, we need to add -2 making it smaller by two orders of magnitude. To compensate, we make the mantissa larger by two orders of magnitude. We have 4.82^3.102-2.
Adjusting the decimal and noting that 100 = 1, we have
4.82^3.100 = 482.3
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After you get the idea, you will be able to do the steps mentally and write down the answer with little effort. Another neat thing about this method is that it can be used to handle engineering notation and SI unit prefix changes. Suppose the result of a calculation is 0.356.10-1 m (meters}. We observe that a more appropriate unit size would be mm (millimeters) which is 10-3 m. We need to make the exponent smaller by 2 to get 10-3, which requires making the mantissa larger by two orders of magnitude to compensate. Thus, we have
0.356.10-1 m = 0.35^6.10-1-2 m = 35.6.10-3 m ←Ans
As you practice, you will have to write down less and less until you can do the changes mentally.