Perform the indicated calculations. Write your answers with...

Perform the indicated calculations. Write your answers with the correct number of significant figures. (a) $\frac{13.01 \mathrm{~g}}{1.23 \mathrm{~L}}$ (b) $17.2 \mathrm{lb}+65.18 \mathrm{lb}-2.3 \mathrm{lb}$ (c) $(0.003240 \mathrm{~cm})(1.25 \mathrm{~cm})\left(1.0 \times 10^{2} \mathrm{~cm}\right)$ (d) $\frac{0.024+\sqrt{(0.024)^{2}+4\left(6.3 \times 10^{-3}\right)}}{2}$ Where 4 and 2 are exact numbers, and all other quantities are measurements. (e) $\frac{\left(4.63 \times 10^{4}\right)\left(1.8 \times 10^{-6}\right)}{\left(6.022 \times 10^{23}\right)}$ Assume all quantities are measurements.

Perform the indicated calculations. Write your answers with the correct number of significant figures.
(a) $\frac{13.01 \mathrm{~g}}{1.23 \mathrm{~L}}$
(b) $17.2 \mathrm{lb}+65.18 \mathrm{lb}-2.3 \mathrm{lb}$
(c) $(0.003240 \mathrm{~cm})(1.25 \mathrm{~cm})\left(1.0 \times 10^{2} \mathrm{~cm}\right)$
(d) $\frac{0.024+\sqrt{(0.024)^{2}+4\left(6.3 \times 10^{-3}\right)}}{2}$
Where 4 and 2 are exact numbers, and all other quantities are measurements.
(e) $\frac{\left(4.63 \times 10^{4}\right)\left(1.8 \times 10^{-6}\right)}{\left(6.022 \times 10^{23}\right)}$
Assume all quantities are measurements.

Key Concepts

Exact Numbers

Exact numbers are values that are known with complete certainty (such as defined constants or counting numbers) and do not limit the precision of the result in significant figure calculations. They do not affect the number of significant figures when used in calculations with measured quantities.

Addition and Subtraction in Significant Figures

For addition and subtraction, the result should be rounded to the same decimal place as the measurement with the least number of decimal places. This ensures that the precision of the result does not exceed that of the least precise measurement.

Scientific Notation

Scientific notation is a way of expressing very large or small numbers in a compact form, where the significant figures are explicitly captured in the coefficient, and the power of ten indicates the magnitude. This format helps in maintaining clarity and consistency in the number of significant digits in calculations.

Significant Figures

Significant figures are the digits in a measured or calculated value that are known with certainty plus one final digit that is an estimate. They indicate the precision of a measurement and play a critical role in ensuring that calculated results do not imply greater precision than the measurements themselves.

Multiplication and Division in Significant Figures

When performing multiplication or division, the final result should be reported using the same number of significant figures as the measurement with the fewest significant figures. This rule reflects the compounded uncertainty from the least precise measurement involved in the calculation.

Transcript

00:01 For this exercise, we'll be solving five problems.

00:05 These five problems will be solved, considering that they're all measurements, unless otherwise specified, and we'll be using significant figures and rounding.

00:14 For problem a, this is an addition, excuse me, it's not, it's a division problem, and we're given 13 .01 grams divided by 1 .23 liters.

00:31 We put this into our calculator and we get 10 .572, which isn't the correct answer.

00:46 Units are grams per liter.

00:48 Since our first value, our numerator has four significant figures and our denominator has three significant figures.

00:58 We go to a different color.

01:00 This value has three significant figures.

01:03 That is our least precise measurement.

01:05 We can report our answer to three significant figures.

01:08 Figures.

01:09 So we're going to round this six, or this five up to a six, since the next number is greater than five.

01:18 Our answer will be 10 .6 grams for liter for the first of our six questions that we're going to do.

01:29 B is adding and subtracting.

01:32 We are given 17 .21 pounds plus 65 .18 .18 pounds.

01:43 Minus 2 .3 pounds.

01:48 When we put these numbers into our calculator, we end up with 80 .09 pounds.

01:53 Again, this is not the correct answer.

01:58 If i write these numbers vertically, 17 .21, i'm going to leave the lean units off for this, 65 .18 and 2 .3.

02:09 We can see that the least number of decimal places is the three right here.

02:17 According to significant figure rules, we can report our value when adding subtracting to the least number of decimal places.

02:26 So our answer will be 80 .1 pounds.

02:36 C is a multiplication, and we're given three values to multiply.

02:42 0 .003240 times 1 .25.

02:52 Those were centimeters.

02:54 Let me go get my units in there.

02:58 1 .25 centimeters and finally 1 .0 times 10 to the second centimeters.

03:13 When we put these numbers into our calculator and i'm going to talk more about that in just a moment, we get the answer of i got 0 .405 cubic centimeters.

03:33 When we look at, i'm going to go into a highlighter here, this value has four significant.

03:45 Figures.

03:47 This has three significant figures and this value has two significant figures.

03:52 Let's write those down quickly.

03:56 This has four significant figures.

04:00 This has three significant figures and this one has two significant figures.

04:05 We can report our values when multiplying and dividing to the number of significant figures in our at least precise measurement, which is two significant figures.

04:20 Rounding this to two significant figures, we get 0 .41 centimeters cubed...

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