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.)

Step by step solution

01

Perform divisions

Calculate each individual division: 1. \(\frac{2.526}{3.1} = 0.8151612903\cdots\) 2. \(\frac{0.470}{0.623} = 0.7544121306\cdots\) 3. \(\frac{80.705}{0.4326} = 186.6513240\cdots\)

02

Determine significant figures for divisions

Consider the number of significant figures in each division: 1. \(2.526\) has 4 sig. figs., and \(3.1\) has 2 sig. figs., so the result should have 2 sig. figs. Round the result of the first division as \(0.82\). 2. \(0.470\) has 3 sig. figs., and \(0.623\) has 3 sig. figs., so the result should have 3 sig. figs. Round the result of the second division as \(0.754\). 3. \(80.705\) has 5 sig. figs., and \(0.4326\) has 4 sig. figs., so the result should have 4 sig. figs. Round the result of the third division as \(186.7\).

03

Perform addition and determine significant figures

Add the rounded results of the divisions and consider significant figures for the addition: \(0.82 + 0.754 + 186.7 = 188.274\) The least number of decimal places among all values is 2 (for the first and third values). Therefore, we should round the result to 2 decimal places: \(a = 188.27\) #b. Calculation of b#

04

Perform multiplications and subtractions

Calculate the multiplication and subtraction operations: 1. \(6.404 \times 2.91 = 18.64364\) 2. \(18.7 - 17.1 = 1.6\)

05

Determine significant figures for multiplication and subtraction

1. \(6.404\) has 4 sig. figs., and \(2.91\) has 3 sig. figs., so the result should have 3 sig. figs. Round the result of multiplication as \(18.6\). 2. Both \(18.7\) and \(17.1\) have 3 sig. figs., so the result should have 1 decimal place. The result of subtraction is \(1.6\).

06

Perform division and determine significant figures

Divide the results of the multiplication and subtraction operations: \(\frac{18.6}{1.6} = 11.625\) Both numerator and denominator have 3 sig. figs., so the result should also have 3 sig. figs. Round the result as: \(b = 11.6\) #c. Calculation of c#

07

Perform subtractions

Subtract the given numbers: \(6.071 \times 10^{-5} - 8.2 \times 10^{-6} - 0.521 \times 10^{-4} = 1.0709 \times 10^{-5}\)

08

Determine significant figures

Each value in the subtraction has 3 sig. figs. (ignoring the powers of 10), so the result should also have 3 sig. figs.: \(c = 1.07 \times 10^{-5}\) #d. Calculation of d#

09

Perform additions and determine significant figures

Add the given numbers and consider significant figures for the addition: 1. \(3.8 \times 10^{-12} + 4.0 \times 10^{-13} = 3.84 \times 10^{-12}\) (2 sig. figs. for each value, so the result should have 2 sig. figs.) 2. \(4 \times 10^{12} + 6.3 \times 10^{13} = 6.34 \times 10^{13}\) (1 and 2 sig. figs. in the values, respectively, so the result should have 2 sig. figs.)

10

Perform division and determine significant figures

Divide the results of the additions: \(\frac{3.84 \times 10^{-12}}{6.34 \times 10^{13}} = 6.059308 \times 10^{-26}\) Since both numerator and denominator have 2 sig. figs., the result should also have 2 sig. figs.: \(d = 6.1 \times 10^{-26}\) #e. Calculation of e#

11

Perform addition and division

Add the given numbers and divide by 4: \(\frac{9.5 + 4.1 + 2.8 + 3.175}{4} = 4.89375\)

12

Determine significant figures

Since the number 4 in the denominator is exact, we will only consider the sig. figs. of the numbers of the numerator. Among those numbers, the least number of sig. figs. is 2, so the result should have 2 sig. figs.: \(e = 4.9\) #f. Calculation of f#

13

Perform subtraction and division

Subtract the given numbers, divide by the result, and multiply by 100: \(\frac{8.925 - 8.905}{8.925} \times 100 = 0.224215247\%)

14

Determine significant figures

Both \(8.925\) and \(8.905\) have 4 sig. figs., so the result of the subtraction should have 4 sig. figs. The denominator has 4 sig. figs., and 100 is an exact number, so the final result should have 4 sig. figs.: \(f = 0.2242\%\)

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