Perform the following mathematical operations and express...

Perform the following mathematical operations and express the result to the correct number of significant figures. a. $\frac{2.526}{3.1}+\frac{0.470}{0.623}+\frac{80.705}{0.4326}$ b. $(6.404 \times 2.91) /(18.7-17.1)$ c. $6.071 \times 10^{-5}-8.2 \times 10^{-6}-0.521 \times 10^{-4}$ d. $\left(3.8 \times 10^{-12}+4.0 \times 10^{-13}\right) /\left(4 \times 10^{12}+6.3 \times 10^{13}\right)$ e. $\frac{9.5+4.1+2.8+3.175}{4}$ (Assume that this operation is taking the average of four numbers. Thus 4 in the denominator is exact.) f. $\frac{8.925-8.905}{8.925} \times 100$ (This type of calculation is done many times in calculating a percentage error. Assume that this example is such a calculation; thus 100 can be considered to be an exact number.)

Perform the following mathematical operations and express the result to the correct number of significant figures.
a. $\frac{2.526}{3.1}+\frac{0.470}{0.623}+\frac{80.705}{0.4326}$
b. $(6.404 \times 2.91) /(18.7-17.1)$
c. $6.071 \times 10^{-5}-8.2 \times 10^{-6}-0.521 \times 10^{-4}$
d. $\left(3.8 \times 10^{-12}+4.0 \times 10^{-13}\right) /\left(4 \times 10^{12}+6.3 \times 10^{13}\right)$
e. $\frac{9.5+4.1+2.8+3.175}{4}$
(Assume that this operation is taking the average of four numbers. Thus 4 in the denominator is exact.)
f. $\frac{8.925-8.905}{8.925} \times 100$
(This type of calculation is done many times in calculating a percentage error. Assume that this example is such a calculation; thus 100 can be considered to be an exact number.)

Key Concepts

Exact Numbers

Exact numbers are values that are known with complete certainty, such as defined constants or counting numbers. They do not affect the number of significant figures in the result, since they have an infinite number of significant figures, ensuring that their presence does not limit the precision of calculations.

Percentage Error Calculation

Percentage error is a way to express how much an observed or measured value deviates from a true or accepted value, usually calculated by taking their difference divided by the accepted value and multiplying by 100. When performing these calculations, significant figures rules are applied, treating the factor 100 as exact.

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a compact form. It represents numbers as a product of a coefficient and a power of 10, which makes it easier to handle calculations and keep track of significant figures and rounding.

Addition and Subtraction Rules

For addition and subtraction, the precision of the result is dictated by the least precise decimal place among the numbers involved, rather than the total number of significant figures. The final answer should be rounded to reflect the highest uncertainty in the measurement’s decimal places.

Multiplication and Division Rules

When performing multiplication or division, the result should be rounded to the same number of significant figures as the measurement with the fewest significant figures used in the calculation. This ensures that the uncertainty in the calculation is not understated or overstated.

Significant Figures

Significant figures are the digits in a number that carry meaningful information about its precision. They include all certain digits plus the first uncertain digit. It is important to express the final answer in the appropriate number of significant figures to reflect the precision of the measured or given values.

Transcript

00:01 In this question we need to do the math and then figure out the product with the right significant number.

00:11 So this is a pretty complicated mass.

00:14 So if you do all the calculation, the product will be 188 .1 .27.

00:25 And so we need to decide the number of 6 .7.

00:30 Significant figure of this in order to give us the right product.

00:34 And in order to do that, let's take a look at each calculation over here.

00:40 So this 2 .5 .16 divided by 3 .1 is addition and, i'm sorry, multiplication and decision and so division.

00:59 And so the number of significant figure of the product will be, the product will contain the same number of significant figure with the number that has the least number of significant figure.

01:14 So in this case, this one has one, two, three, four significant figure.

01:18 This one has only two, so the product will contain only two significant figure.

01:23 That's why the right product will be 0 .81 plus same.

01:29 Them roll three significant figure, three significant figures, so 0 .754.

01:36 This one has one, two, three, four, five significant figure, four significant figure.

01:42 It's going to be 186 .6.

01:45 Now, this is between addition.

01:49 So the product must contain the least decimal place, and this one is only one place and that's why the right product would be 188 .1 because we route up to here.

02:08 B is more simple mass so we can do just straight from here.

02:17 So the first one over here, 6404 times 2 .91 is multiplication.

02:27 And so that's why the product must have.

02:30 Have the same significant number as the number has the least significant number...

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