Perform the following mathematical operations, and express...

Perform the following mathematical operations, and express the result to the correct number of significant figures. a. $6.022 \times 10^{23} \times 1.05 \times 10^{2}$ b. $\frac{6.6262 \times 10^{-34} \times 2.998 \times 10^{8}}{2.54 \times 10^{-9}}$ c. $1.285 \times 10^{-2}+1.24 \times 10^{-3}+1.879 \times 10^{-1}$ d. $\frac{(1.00866-1.00728)}{6.02205 \times 10^{2.3}}$ e. $\frac{9.875 \times 10^{2}-9.795 \times 10^{2}}{9.875 \times 10^{2}} \times 100(100$ is exact) f. $\frac{9.42 \times 10^{2}+8.234 \times 10^{2}+1.625 \times 10^{3}}{3}(3$ is exact)

Perform the following mathematical operations, and express the result to the correct number of significant figures.
a. $6.022 \times 10^{23} \times 1.05 \times 10^{2}$
b. $\frac{6.6262 \times 10^{-34} \times 2.998 \times 10^{8}}{2.54 \times 10^{-9}}$
c. $1.285 \times 10^{-2}+1.24 \times 10^{-3}+1.879 \times 10^{-1}$
d. $\frac{(1.00866-1.00728)}{6.02205 \times 10^{2.3}}$
e. $\frac{9.875 \times 10^{2}-9.795 \times 10^{2}}{9.875 \times 10^{2}} \times 100(100$ is exact)
f. $\frac{9.42 \times 10^{2}+8.234 \times 10^{2}+1.625 \times 10^{3}}{3}(3$ is exact)

Key Concepts

Addition and Subtraction

For addition and subtraction, the result should be rounded to the same decimal place as the measurement with the fewest decimal places. This adjustment avoids implying a false level of precision in the final result.

Exact Numbers

Exact numbers, such as counts or defined constants given without uncertainty, are considered to have an infinite number of significant figures, meaning they do not limit the precision of a calculation. They are not used to determine the number of significant figures in the final answer.

Significant Figures

Significant figures represent the digits in a measurement that carry meaningful contributions to its precision. They are essential in scientific calculations because they prevent the overstatement of precision beyond the accuracy of the original data.

Multiplication and Division

In multiplication or division operations, the result should be reported using the same number of significant figures as the factor with the fewest significant figures. This rule ensures that the precision of the computed value reflects the least precise measurement involved in the calculation.

Scientific Notation

Scientific notation is a method of expressing large or small numbers in a compact form, using a coefficient and a power of ten. It is particularly useful in significant figures discussion because the coefficient directly shows the significant digits, making it easier to apply rounding rules appropriately.

Transcript

00:01 So we have given here some mathematical operations.

00:03 First part is 6 .022 multiplying 10 raised power 23 multiplying 1 .05 multiplying 10 raised power 2.

00:27 So this will be solved as 6 .022 multiplying 10 % power 2.

00:35 So see here the powers, so they will be taken on one side, okay? and this 6 .022 will be multiplied by 1 .05 as usual we do.

00:49 And then we have after for solving these powers, we have 10 rest power 2 here, okay, 23.

00:57 So the powers will be add, okay, in the case of multiplication, the powers will be add.

01:04 And this will be 10 rest of power 23 plus 2.

01:08 Okay, so this will be 6 .323 multiplying 10 raised to power 25.

01:19 Now we have b part 6 .6 .62 multiplying 10 raised to power minus 34 multiplying 10 % 2 .998, multiplying 10 raise to power 8 divided by 2 .54, multiplying 10 raise to power minus 9.

01:59 So, first of all, multiplying, so this will be here 6 .6262, multiplying, multiplying 2 .962, multiplying 2 .98.

02:21 Multiplying 10 raised power minus 34 plus 8, okay, during multiplication powers will add.

02:32 Okay, and now this 10 raised power minus 9, when this power goes to numerator side, so the power will be positive, okay? so this can be written as plus 9 divided by 2 .54.

02:49 So now, multiplying 6 .6 .6.

02:52 262 by 2 .998 and then dividing the product by 2 .54 and solving the powers we will get.

03:05 So this will be 19 .8653476 divided by 2 .544.

03:18 Multiplying solving the powers so we have 10 rift power minus 17 and this will be 7 .821 so i'm here this is we got after rounding of the figures okay so then multiplying 10 risk to power minus 17 so this will be the answer c part we have 1 .285 multiplying 10 raise to power minus 2 plus 1 .24 multiplying 10 raise to power minus 3 plus 1 .879 multiplying 10 raise to power minus 1.

04:27 So now when we have the figures in addition, okay? so in that case, we will move the decimals such that the powers will become same.

04:44 So now this can be written as that's equal to 1 .285 multiplying 10 raise to power minus 2 plus.

04:58 Now shifting the decimal to left side, so power will become positive and this will be 0 .124 multiplying 10 raise to power minus 2 plus.

05:13 Now here shifting the decimal to right side, this will be 18 .79 multiplying 10 raise to power minus 2.

05:23 So now you can see that the powers are same right.

05:25 So now we will take the power as common.

05:30 So this will be 1 .285 plus 0 .124 plus 18 .79 multiplying 10 rest power minus 2.

05:50 So this will be this will be 20 .19 multiplying 10 rest power minus 2.

05:55 So this will be 20 .19 multiplying 10 rest power minus 2.

05:58 2.

06:00 Now, next part, d part.

06:03 So this time we have 1 .285 multiplying 10 raise to power minus 2 minus 1 .24 multiplying 10 raise to power minus 3.

06:26 Okay.

06:28 So in the case of subtraction also, we will make the power same so that we can take the power as common okay so this will be equals to 1 .285 multiplying 10 raise to power minus 2 minus shifting the decimal to left side so the power will be positive and this will be 0 .124 multiplying 10 raise to power minus 2 okay so now this can be written as 1 .1 .2.

07:04 28 minus 0 .124 multiplying 10 rift to power minus 2...

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