Round the answer for each problem to the correct number of...

Round the answer for each problem to the correct number of significant figures. a. $\left(7.31 \times 10^{4}\right)+\left(3.23 \times 10^{3}\right)$ b. $\left(8.54 \times 10^{-3}\right)-\left(3.41 \times 10^{-4}\right)$ c. $4.35 \mathrm{dm} \times 2.34 \mathrm{dm} \times 7.35 \mathrm{dm}$ d. $4.78 \mathrm{cm}+3.218 \mathrm{cm}+5.82 \mathrm{cm}$ e. $38,736 \mathrm{km} \div 4784 \mathrm{km}$

Round the answer for each problem to the correct number of significant figures.
a. $\left(7.31 \times 10^{4}\right)+\left(3.23 \times 10^{3}\right)$
b. $\left(8.54 \times 10^{-3}\right)-\left(3.41 \times 10^{-4}\right)$
c. $4.35 \mathrm{dm} \times 2.34 \mathrm{dm} \times 7.35 \mathrm{dm}$
d. $4.78 \mathrm{cm}+3.218 \mathrm{cm}+5.82 \mathrm{cm}$
e. $38,736 \mathrm{km} \div 4784 \mathrm{km}$

Key Concepts

Addition and Subtraction Rules

When adding or subtracting numbers, the result should be rounded to the least number of decimal places present in any of the original numbers. This rule ensures that the precision of the result is not overstated compared to the measurements involved.

Scientific Notation

Scientific notation is a method of writing very large or very small numbers in the form of a coefficient multiplied by a power of ten. This notation simplifies arithmetic operations and clearly shows the number of significant figures in the coefficient.

Multiplication and Division Rules

For multiplication and division, the final result should have the same number of significant figures as the number with the fewest significant figures used in the calculation. This rule preserves the overall precision of the computed quantity.

Significant Figures

Significant figures define the precision of a number by showing which digits are known reliably. In measurement and calculations, only these digits are considered trustworthy, and they help indicate the uncertainty in reported values.

Rounding

Rounding is the process of reducing the number of digits in a numeral while maintaining its value as close as possible to the original. In the context of significant figures, rounding involves adjusting the number to reflect the precision of the measurement or calculation.

Transcript

00:01 Okay, so this problem wants you to round the answer for each problem to the correct number of significant figures.

00:10 So it's really important for these problems is that you do want to solve the problem before you round.

00:18 Don't make the mistake of rounding before you solve the problem because your answer might not be as precise.

00:26 So if you're looking, let's look at part a.

00:28 So anytime you add together values that are written in scientific notation, you need to convert them so you have the exponents the same.

00:41 And so you always want to convert the lower exponent to a higher exponent.

00:46 So in this case, you would move the decimal point one place to the left.

00:53 And so you would have 0 .323 times 10 to the third.

01:02 And if you add these values together, without rounding, you get 7 .633 times 10 to the power of 4.

01:19 So anytime you're adding values, you want to look at how many decimal points are to the right of your, how many places you have to the right of your decimal point.

01:41 So 7 .31, you have one place to the right of your decimal point, and for 3 .23 .3 .3, you have 3 to the right of your decimal point.

01:53 So you want to round this to two places to the right of your decimal point.

02:00 So if you were to round this, because this three after your last significant figure of three doesn't round it up, so your answer would be 7 .63 times 10 to the 4.

02:23 And so you have a similar rule when you do subtraction.

02:29 So again, the first thing you want to do is make sure that you have your exponent the same.

02:37 So you'd have 0 .341 times 10 to the negative 3.

02:45 And then if you subtract these values, you get 8 .199, 8 .199 times 10 to the negative 3.

03:00 And now you have two values, two numerals to the right of the decimal point.

03:06 Here you have three because i forgot the one.

03:10 So you want to have two values to the right of the decimal point.

03:16 So now this one's a little tricky because you have a nine after a nine.

03:22 This rounds this value from nine up, but because and you have to actually round the next decimal place up as well.

03:33 So you have a value of 8 .20 times 10, oops, times 10, to the negative 3.

03:45 And you want to make sure that you do include the 0 because this 0, because it's to the right of the decimal point, is a significant figure that needs to be accounted for.

03:59 Okay, so let's look at part c...

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