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

a. Division and Addition

Perform the divisions first and round the obtained values according to the lowest significant figures in the divisor and dividend: \[\frac{2.526}{3.1} = 0.815\] \[\frac{0.470}{0.623} = 0.754\] \[\frac{80.705}{0.4326} = 186.7\] Now, sum these values and round the result according to the lowest decimal place: \[0.815 + 0.754 + 186.7 = 188.268 \approx 188.3\]

02

b. Multiplication, Subtraction, and Division

Perform the multiplication first and round the obtained value according to the lowest significant figures in the multiplicands: \[6.404 \times 2.91 = 18.63464 \approx 18.6\] Next, subtract the two numbers and round the result according to the lowest decimal place: \[18.7 - 17.1 = 1.6\] Finally, divide the two values and round the result according to the lowest significant figures in the divisor and dividend: \[\frac{18.6}{1.6}= 11.625 \approx 11.6\]

03

c. Subtraction

Perform the subtractions by considering the least amount of decimal places in the subtraction and rounding the results accordingly: \[6.071 \times 10^{-5} - 8.2 \times 10^{-6} = 5.289 \times 10^{-5}\] \[5.289 \times 10^{-5} - 0.521 \times 10^{-4} = 4.768 \times 10^{-5}\]

04

d. Addition and Division

Add the numbers in the numerator and denominator. Keep the lowest amount of significant figures while adding: \[(3.8 \times 10^{-12} + 4.0 \times 10^{-13}) = 3.84 \times 10^{-12}\] \[(4 \times 10^{12}+6.3 \times 10^{13}) = 6.64 \times 10^{13}\] Now, divide the two values and round the result according to the lowest significant figures in the divisor and dividend: \[\frac{3.84 \times 10^{-12}}{6.64 \times 10^{13}} = 5.78 \times 10^{-26}\]

05

e. Addition and Division

Perform the addition first as it is an average of four numbers, the four in the denominator is an exact number: \[9.5 + 4.1 + 2.8 + 3.175 = 19.575\] Now divide the sum by 4 and round to the lowest decimal place: \[\frac{19.575}{4} = 4.89375 \approx 4.894\]

06

f. Subtraction, Division, and Multiplication

Perform the subtraction first and keep the lowest amount of decimal places: \[8.925 - 8.905 = 0.020\] Now divide the result by the given value and round according to the lowest significant figures in the divisor and dividend: \[\frac{0.020}{8.925} = 0.0022402\] Finally, multiply the result by 100 (since it's an exact number): \[0.0022402 \times 100 = 0.22402 \approx 0.224\% \]

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