U Can: Chemistry I For Dummies. Hren Chris

yields a coefficient quotient of

, and dividing the powers of 10 (by subtracting their exponents) yields a quotient of

. Marrying the two quotients produces the given answer, already in scientific notation.

      3. 1.82. First, convert each number to scientific notation:

and

. Next, multiply the coefficients:

. Then add the exponents on the powers of 10:

. Finally, join the new coefficient with the new power:

. Expressed in scientific notation, this answer is

. (Note: Looking back at the original numbers, you see that both factors have only two significant figures; therefore, you should round your answer to match that number of sig figs, making it 1.8. See the later sections “Identifying Significant Figures” and “Doing Arithmetic with Significant Figures” for details.)

      4.

. First, convert each number to scientific notation:

and

. Then divide the coefficients:

. Next, subtract the exponent in the denominator from the exponent in the numerator to get the new power of 10:

. Join the new coefficient with the new power:

. Finally, express gratitude that the answer is already conveniently expressed in scientific notation.

      Using Scientific Notation to Add and Subtract

      Addition or subtraction gets easier when you express your numbers as coefficients of identical powers of 10. To wrestle your numbers into this form, you may need to use coefficients less than 1 or greater than 10. So scientific notation is a bit too strict for addition and subtraction, but exponential notation still serves you well.

       Remember: To add two numbers easily by using exponential notation, first express each number as a coefficient and a power of 10, making sure that 10 has the same exponent in each number. Then add the coefficients. To subtract numbers in exponential notation, follow the same steps but subtract the coefficients.

       Examples

      Q. Use exponential notation to add these numbers:

.

      A.

. First, write both numbers with the same power of 10:

      Next, add the coefficients:

      Finally, join your new coefficient to the shared power of 10:

      Q. Use exponential notation to subtract:

.

      A.

. First, convert both numbers to the same power of 10. We’ve chosen 10^– 2:

      Next, subtract the coefficients:

      Then join your new coefficient to the shared power of 10:

       Practice Questions

      1. Add

.

      2. Subtract

.

      3. Use exponential notation to add

.

      4. Use exponential notation to subtract

.

       Practice Answers

      1.

. Because the numbers are each already expressed with identical powers of 10, you can simply add the coefficients:

. Then join the new coefficient with the original power of 10.

      2.

. Because the numbers are each expressed with the same power of 10, you can simply subtract the coefficients:

. Then join the new coefficient with the original power of 10.

      3.

(or an equivalent expression). First, convert the numbers so they each use the same power of 10:

and

. Here, we use 10^– 3, but you can use a different power as long as the power is the same for each number. Next, add the coefficients:

. Finally, join the new coefficient with the shared power of 10.

      4.

(or an equivalent expression). First, convert the numbers so each uses the same power of 10:

and

. Here, we’ve picked 10², but any power is fine as long as the two numbers have the same power. Then subtract the coefficients:

. Finally, join the new coefficient with the shared power of 10.

      Distinguishing between Accuracy and Precision

      Whenever you make measurements, you must consider two factors, accuracy and precision. Accuracy is how well the measurement agrees with the accepted or true value. Precision is how well a set of measurements agree with each other. In chemistry, measurements should be reproducible; that is, they must have a high degree of precision. Most of the time chemists make several measurements and average them. The closer these measurements are to each other, the more confidence chemists have in their measurements. Of course, you also want the measurements to be accurate, very close to the correct answer. However, many times you don’t know beforehand anything about the correct answer; therefore, you have to rely on precision as your guide.

      Suppose you ask four lab students to make three measurements of the length of the same object. Their data follows:

      The accepted length of the object is 27.55 cm. Which of these students deserves the higher lab grade? Both students 1 and 3 have values close to the accepted value, if you just consider their average values. (The average, found by summing the individual measurements and dividing by the number of measurements, is normally considered to be more useful than any individual value.) Both students 1 and 3 have made accurate determinations of the length of the object. The average values determined by students 2 and 4 are not very close to the accepted value, so their values are not considered to be accurate.

      However,