Perform the following mathematical operations, and express...

Perform the following mathematical operations, and express each result to the correct number of significant figures. a. $\frac{0.102 \times 0.0821 \times 273}{1.01}$ b. $0.14 \times 6.022 \times 10^{23}$ c. $4.0 \times 10^{4} \times 5.021 \times 10^{-3} \times 7.34993 \times 10^{2}$ d. $\frac{2.00 \times 10^{6}}{3.00 \times 10^{-7}}$

Perform the following mathematical operations, and express each result to the correct number of significant figures.
a. $\frac{0.102 \times 0.0821 \times 273}{1.01}$
b. $0.14 \times 6.022 \times 10^{23}$
c. $4.0 \times 10^{4} \times 5.021 \times 10^{-3} \times 7.34993 \times 10^{2}$
d. $\frac{2.00 \times 10^{6}}{3.00 \times 10^{-7}}$

Key Concepts

Scientific Notation

Scientific notation expresses numbers as a product of a coefficient and a power of ten, making it easier to work with very large or very small numbers. It also aids in clearly indicating and preserving the number of significant figures during arithmetic operations, especially in multiplication and division.

Rounding Techniques

Rounding techniques are used to limit the number of significant figures in a number, adjusting the final digit based on the value of the next digit. Proper rounding is crucial in order to correctly express the numerical results according to the rules of significant figures without overstating the precision.

Significant Figures

Significant figures refer to the digits in a number that carry meaning in terms of its measurement resolution. They indicate the precision of the measured or calculated quantity, and the rules of significant figures help to ensure that the reported result does not imply a higher level of precision than the original data provides.

Multiplication and Division Operations

For multiplication and division, the key rule is that the final result must be rounded to the same number of significant figures as the measure with the fewest significant figures. This ensures that the precision of the calculated result reflects the least precise measurement used in the operation.

Transcript

00:01 So this question is not only trying to remind you of your order of operations, but also, uh, being able to figure out how many when you have these long chains off multiplication division with a lot of interesting numbers.

00:16 What's the, um what's the number of significant figures that you have to, uh, putting your final answer right? so therefore, parts, luckily for us, it's all multiplication.

00:27 There's some scientific notation, but let's just get started, right? so i had written out each of the problems on each whiteboard tab.

00:34 Um right, so for this problem, it's just multiplication division.

00:37 So 0.102 times 0.821 times 273 over 1.1 when you put all that into your calculator, right, um, you do all the multiplication on top what i did first and then divide by 1.1 so all the numbers here have three significant figures 30.10 to all of those are significant.

00:59 8 to 1 in 10.8 to 1 or significant, and then to 73 are all significant.

01:04 1.1 they're all three significant figures.

01:07 So when you get the answer.

01:09 Um, i got 2.264 and then you can round that 23 significant figures.

01:17 So that'll be 2.26 as your final answer.

01:22 Part b.

01:22 Um, we have our first example of scientific notation.

01:25 Right.

01:25 So, um, in terms of significant figures 0.14 the one of the four a significant you have four significant figures here.

01:33 Um, so our final answer is only gonna have to.

01:36 So when you do that multiplication, you get, uh, eight points, 43 0/8 times 10 to the 20 seconds, and then that's only gonna have two significant figures.

01:51 So you can round that to 8.4 times 10 to the 20 seconds for this part.

01:59 It's all scientific notation in terms of significant figures.

02:03 It gets more difficult when you have to add a bunch of different scientific notations where you have different exponents.

02:08 But in our case, a cz long as you pulled into the calculator, right, um, you should be okay, right? so when shows no significant figures, um, we're gonna have to round two to because 4.0 has the fewest number of significant figures.

02:21 Therefore here...

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