Question

Evaluate each of the following, and write the answer to the appropriate number of significant figures. a. \(\left(2.9932 \times 10^{4}\right)\left[2.4443 \times 10^{2}+1.0032 \times 10^{1}\right]\) b. \(\left[2.34 \times 10^{2}+2.443 \times 10^{-1}\right] /(0.0323)\) c. \(\left(4.38 \times 10^{-3}\right)^{2}\) d. \(\left(5.9938 \times 10^{-6}\right)^{1 / 2}\)

Short answer

a. \( 7.609 \times 10^6 \) b. \( 7.25 \times 10^3 \) c. \( 1.92 \times 10^{-5} \) d. \( 2.4488 \times 10^{-3} \)

Step-by-step solution

a. Multiply 2.9932 * 10^4 and add the two numbers in the square brackets, then evaluate.

First, we will evaluate the expression inside the square brackets: \(2.4443 \times 10^{2}+1.0032 \times 10^{1}.\) \( 2.444 \times 10^2 = 244.3 \) \( 1.0032 \times 10^1 = 10.03 \) Now, add these two numbers: \( 244.3 + 10.03 = 254.33 \) Next, multiply the result by \(2.9932 \times 10^{4}\): \( (2.9932 \times 10^4)(254.33) \) \( 2.9932 \times 10^4 = 29932 \) \( 29932 \times 254.33 = 7.60912716 \times 10^6 \) Finally, we need to round the answer to the appropriate number of significant figures, which is the lowest number of significant figures in each part (4 significant figures): \( 7.609 \times 10^6 \) Answer: \( 7.609 \times 10^6 \)

b. Add the two numbers in the square brackets, then divide by 0.0323.

First, add the numbers inside the square brackets: \(2.34 \times 10^{2}+2.443 \times 10^{-1}\) \( 2.34 \times 10^2 = 234 \) \( 2.443 \times 10^{-1} = 0.244 \) Now, add these two numbers: \( 234 + 0.244 = 234.244 \) Next, divide the result by 0.0323: \( 234.244 / 0.0323 = 7,250.09669 \) Now, round the answer to the appropriate number of significant figures (3 significant figures): \( 7.25 \times 10^3 \) Answer: \( 7.25 \times 10^3 \)

c. Square (4.38 * 10^-3).

Square the number: \( (4.38 \times 10^{-3})^2 = (4.38)^2 \times (10^{-3})^2 \) \( (4.38)^2 = 19.1844 \) \( (10^{-3})^2 = 10^{-6} \) Now, multiply the two results: \( 19.1844 \times 10^{-6} = 1.91844 \times 10^{-5} \) Finally, round the result to the appropriate number of significant figures (3 significant figures): \( 1.92 \times 10^{-5} \) Answer: \( 1.92 \times 10^{-5} \)

d. Take the square root of (5.9938 * 10^-6).

Take the square root of the number: \( \( \sqrt{5.9938 \times 10^{-6}} = \sqrt{5.9938} \times \sqrt{10^{-6}} \) \) \( \sqrt{5.9938} = 2.448755 \) \( \sqrt{10^{-6}} = 10^{-3} \) Now, multiply the two results: \( 2.448755 \times 10^{-3} = 2.448755 \times 10^{-3} \) Finally, round the result to the appropriate number of significant figures (5 significant figures): \( 2.4488 \times 10^{-3} \) Answer: \( 2.4488 \times 10^{-3} \)