Question

Evaluate each of the following and write the answer to the appropriate number of significant figures. a. \((2.0944+0.0003233+12.22) /(7.001)\) b. \(\left(1.42 \times 10^{2}+1.021 \times 10^{3}\right) /\left(3.1 \times 10^{-1}\right)\) c. \(\left(9.762 \times 10^{-3}\right) /\left(1.43 \times 10^{2}+4.51 \times 10^{1}\right)\) d. \(\left(6.1982 \times 10^{-4}\right)^{2}\)

Short answer

a. \(2.089\) b. \(3.34\times10^5\) c. \(5.18\times10^{-5}\) d. \(3.8415\times10^{-7}\)

Step-by-step solution

a. Addition and division with significant figures

Given expression: \((2.0944+0.0003233+12.22) /(7.001)\) Step 1: Addition Add the numbers: \(2.0944 + 0.0003233 + 12.22 = 14.6346233\) Now, apply the rules of significant figures for addition. In this case, the lowest number of decimal places is 2, so we should have 2 decimal places in our answer: \(14.63\) Step 2: Division Divide the sum by \(7.001\): \(\frac{14.63}{7.001} = 2.0892732\ldots\) Now, apply the significant figure rules for division. In this case, we must have the same number of significant figures as the lowest number of significant figures in the original numbers, which is 4. So our answer should have 4 significant figures: \(2.089\) Therefore, the final answer is \(2.089\).

b. Addition in scientific notation and division with significant figures

Given expression: $\frac{1.42 \times 10^{2}+1.021 \times 10^{3}}{3.1 \times 10^{-1}}$ Step 1: Addition in scientific notation First, express the numbers with the same exponent: \(1.42 \times 10^{2} = 0.0142 \times 10^{4}\) Now, add the given numbers: \(0.0142 \times 10^{4} + 1.021 \times 10^{4} = 1.0352 \times 10^{4}\) Step 2: Division Divide the sum by \(3.1 \times 10^{-1}\): \(\frac{1.0352 \times 10^{4}}{3.1 \times 10^{-1}} = 3.341935\ldots \times 10^{5}\) Now, apply the significant figure rules for division. In this case, we must have the same number of significant figures as the lowest number of significant figures in the original numbers, which is 3. So our answer should have 3 significant figures: \(3.34\times10^5\) Therefore, the final answer is \(3.34\times10^5\).

c. Division in scientific notation with significant figures

Given expression: \(\frac{9.762 \times 10^{-3}}{1.43 \times 10^{2}+4.51 \times 10^{1}}\) Step 1: Division Divide the given numbers: First, resolve the denominator sum: \(1.43 \times 10^2 + 4.51 \times 10^1 = 0.0143 \times 10^{4} + 0.451 \times 10^{3} = 1.883 \times 10^{2}\) Now, divide the numbers: \(\frac{9.762 \times 10^{-3}}{1.883 \times 10^{2}} = 5.17802\ldots \times 10^{-5}\) Step 2: Apply significant figures rules The lowest number of significant figures in the original numbers is 3, so our answer should have 3 significant figures: \(5.18\times10^{-5}\) Therefore, the final answer is \(5.18\times10^{-5}\).

d. Power in scientific notation with significant figures

Given expression: \(\left(6.1982 \times 10^{-4}\right)^{2}\) Step 1: Power in scientific notation Square the given number: \((6.1982 \times 10^{-4})^2 = 38.414776\ldots \times 10^{-8}\) Step 2: Apply significant figures rules The original number has 5 significant figures, so our answer should also have 5 significant figures: \(3.8415\times10^{-7}\) Therefore, the final answer is \(3.8415\times10^{-7}\).