Question

14b (unit 1)
b. perform the following operations. find the calculator answer, then round your answer to reflect the correct number of sig figs. express your answers in scientific notation.
calculator answer rounded answer
i. 2.8 × 10⁻⁴ / 6.32 × 10⁻⁵
ii. (3.02 × 10⁹) × (5.9 × 10⁴)

c. perform the following operations, first giving the calculator answer, then rounding to the correct number of sig figs. use the correct units. (for example, m times m equals m², ft − ft = ft, g ÷ ml = g/ml)
calculator answer rounded answer
i. 56 m × 19.3 m
ii. 0.13 s × 90 s
iii. 76.33 ft − 9 ft
iv. 4.73 g ÷ 19.71 ml

d. peter measured the density of iron to be 8.25 g/cm³. the actual density of iron is 7.87 g/cm³. what was his percent error expressed to the correct number of significant figures.

6 • scientific notation
scientific notation regular notation
1.230 × 10⁻⁸
980,100,000
4.302 × 10⁹
0.000004506
2.204 × 10¹³
0.0000304

Explanation:

Part b

i. \( 2.8 \times 10^{-3} \cdot 6.32 \times 10^{-5} \)

Step1: Multiply coefficients and exponents

To multiply numbers in scientific notation, we multiply the coefficients (\(a \times b\)) and add the exponents of 10 (\(10^m \times 10^n = 10^{m + n}\)).
So, \( (2.8 \times 6.32) \times 10^{-3 + (-5)} \)
\( 2.8 \times 6.32 = 17.696 \) and \( -3 + (-5) = -8 \), so we have \( 17.696 \times 10^{-8} \)

Step2: Adjust to proper scientific notation

We need to have one non - zero digit before the decimal point. So, \( 17.696 \times 10^{-8}=1.7696\times 10^{1}\times 10^{-8}=1.7696\times 10^{-7}\) (this is the calculator answer).

Step3: Round to significant figures

The number \(2.8\) has 2 significant figures and \(6.32\) has 3 significant figures. When multiplying, we round to the least number of significant figures, which is 2.
So, \(1.7696\times 10^{-7}\approx1.8\times 10^{-7}\) (rounded answer)

ii. \( (3.02 \times 10^{9}) \times (5.9 \times 10^{4}) \)

Step1: Multiply coefficients and exponents

Multiply the coefficients: \(3.02\times5.9 = 17.818\)
Add the exponents of 10: \(10^{9}\times10^{4}=10^{9 + 4}=10^{13}\)
So, the product is \(17.818\times 10^{13}\)

Step2: Adjust to proper scientific notation

\(17.818\times 10^{13}=1.7818\times 10^{1}\times 10^{13}=1.7818\times 10^{14}\) (calculator answer)

Step3: Round to significant figures

\(3.02\) has 3 significant figures and \(5.9\) has 2 significant figures. We round to 2 significant figures.
\(1.7818\times 10^{14}\approx1.8\times 10^{14}\) (rounded answer)

Part c

i. \( 56\space m\times19.3\space m \)

Step1: Multiply the numbers

\(56\times19.3 = 1080.8\space m^{2}\) (calculator answer)

Step2: Round to significant figures

\(56\) has 2 significant figures and \(19.3\) has 3 significant figures. We round to 2 significant figures.
\(1080.8\space m^{2}\approx1.1\times 10^{3}\space m^{2}\) or \(1100\space m^{2}\) (rounded answer)

ii. \( 0.13\space s\times90\space s \)

Answer:

We move the decimal point 5 places to the right to get \(a = 3.04\) and \(n=-5\)
So, \(0.0000304=3.04\times 10^{-5}\)

Final Answers

Part b

i. Calculator Answer: \(1.7696\times 10^{-7}\); Rounded Answer: \(1.8\times 10^{-7}\)
ii. Calculator Answer: \(1.7818\times 10^{14}\); Rounded Answer: \(1.8\times 10^{14}\)

Part c

i. Calculator Answer: \(1080.8\space m^{2}\); Rounded Answer: \(1.1\times 10^{3}\space m^{2}\) (or \(1100\space m^{2}\))
ii. Calculator Answer: \(11.7\space s^{2}\); Rounded Answer: \(12\space s^{2}\) (or \(1\times 10^{1}\space s^{2}\) depending on sig fig interpretation of 90)
iii. Calculator Answer: \(67.33\space ft\); Rounded Answer: \(67\space ft\)
iv. Calculator Answer: \(0.23997\space g/mL\); Rounded Answer: \(0.240\space g/mL\)

Part d

Percent Error: \(4.83\%\)

Part 6

Scientific Notation	Regular Notation
\(9.801\times 10^{8}\)	\(980,100,000\)
\(4.302\times 10^{9}\)	\(4302000000\)
\(4.506\times 10^{-6}\)	\(0.000004506\)
\(2.204\times 10^{13}\)	\(22040000000000\)
\(3.04\times 10^{-5}\)	\(0.0000304\)