Multiplying with scientific notation

The units of moles in the numerator of the first term and the denominator of the second term cancel, so you re left with the final answer of 4.9x 10 molecules. (If you need tips on multiplying in scientific notation, check out Chapter 1.)... 

To make working with such extreme numbers easier, chemists turn to scientific notation, which is a special kind of exponential notation. Exponential notation simply means writing a number in a way that includes exponents. In scientific notation, every number is written as the product of two numbers, a coefficient and a power of 10. In plain old exponential notation, a coefficient can be any value of a number multiplied by a power with a base of 10 (such as 10" ). But scientists have rules for coefficients in scientific notation. In scientific notation, the coefficient is always at least 1 and always less than 10. For example, the coefficient could be 7, 3.48, or 6.0001. 

To get around this problem we can use scientific notation to write such numbers as a signed decimal number, usually with magnitude greater than or equal to 1 and less than 10, multiplied by an appropriate power of 10. 

For very large numbers with units of measure, use scientific notation or choose an appropriate multiplying prefix for the unit to avoid numbers of more than four digits. 

When multiplying numbers in scientific notation, what do you do with the exponents ... 

Choice b is correct. You start with 3,600,000 and then you count over from the current decimal point until you find where you will put your decimal point. In this case you count over 6 places, because your goal is to put a new decimal point after the 3 and before the 6. This is because scientific notation dictates that the answer be in the form 3.6 times a power of 10. Since you counted over 6 places, you will multiply 3.6 by 106. This makes sense because 10(- = 1,000,000, and 3.6 x 1,000,000 = 3,600,000. 

With very large or very small numbers, to avoid many zeros that can be easily misread, use prefixes for units (e.g., pM, nm, MPa) or use scientific notation by multiplying by 10 to a suitable power as we have done here. 

Strategy Let s express scientific notation as TV x lo . When multiplying numbers using scientific notation, multiply the TV parts of the numbers in the usual way. To come up with the correct exponent, n, we add the exponents. 

Write each of the following numbers in standard scientific notation. See the Appendix if you need help multiplying or dividing numbers with exponents. 

If you are familiar with scientific nofa-fion, you will see fhaf fhe following statemenfs are frue. To convert from % to ppm or ppb, simply multiply by 10" or 10, respectively. To convert from ppb fo ppm or %, simply divide by lO" or 10. respectively. To review the use of scientific notation, see Appendix B. 

Scientific notation, in which large and small numbers are represented by a number between 1 and multiplied by 10 with an exponent, is reviewed in Appendix B. 

Scientific notation (see Appendix lA) allows us to express very large or very small quantities in a compact manner by using exponents. For example, the diameter of a hydrogen atom can be written as 1.06 x 10 m. The International System of Units uses the prefix multipliers shown in Table 1.2 with the standard units. These multipliers change the value of the unit by powers of 10 (just like an exponent does in scientific notation). For example, the kilometer has the prefix kilo meaning 1000 or 10. Therefore,... 

Use scientific notation to express each quantity with only the base units (no prefix multipliers). 

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