Question

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\trounded answer
i. ( 2.8 \times 10^3 / 6.32 \times 10^5 )\t\t
ii. ( (3.02 \times 10^3) \times (5.9 \times 10^4) )\t\t

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 ( \text{m}^2 ); ft. – ft. = ft.; g ÷ ml = g/ml)

calculator answer\trounded answer
i. ( 56 , \text{m} \times 19.3 , \text{m} )\t\t
ii. ( 0.13 , \text{s} \times 90 , \text{s} )\t\t
iii. ( 76.33 , \text{ft} - 9 , \text{ft} )\t\t
iv. ( 4.73 , \text{g} div 19.71 , \text{ml} )\t\t

d. peter measured the density of iron to be ( 8.25 , \text{g/cm}^3 ). the actual density of iron is ( 7.87 , \text{g/cm}^3 ). what was his percent error expressed to the correct number of significant figures.

6 • scientific notation

scientific notation	regular notation
	( 980,100,000 )
( 4.302 \times 10^9 )
	( 0.000004506 )
( 2.204 \times 10^{13} )
	( 0.0000304 )

Explanation:

Part b

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

Step 1: Divide the coefficients and subtract exponents

To divide numbers in scientific notation, we divide the coefficients (\(a\) and \(c\)) and subtract the exponents (\(b\) and \(d\)) using the rule \(\frac{a\times10^{b}}{c\times10^{d}}=\frac{a}{c}\times10^{b - d}\).
For \(2.8\times10^{-3}\div6.32\times10^{5}\), the coefficient division is \(\frac{2.8}{6.32}\approx0.4430\) and the exponent subtraction is \(- 3-5=-8\). So the calculator answer is \(0.4430\times10^{-8}\). But we need to adjust to proper scientific notation (coefficient between 1 and 10). So we rewrite \(0.4430\times10^{-8}\) as \(4.430\times10^{-9}\).

Step 2: Round to significant figures

The original numbers \(2.8\) (2 sig figs) and \(6.32\) (3 sig figs). When dividing, the result should have the least number of sig figs, which is 2. So we round \(4.430\times10^{-9}\) to \(4.4\times10^{-9}\).

Calculator Answer: \(4.430\times10^{-9}\)

Rounded Answer: \(4.4\times10^{-9}\)

ii. \( (3.02\times 10^{3})\times(5.9\times 10^{4}) \)

Step 1: Multiply the coefficients and add exponents

Using the rule \((a\times10^{b})\times(c\times10^{d})=(a\times c)\times10^{b + d}\). The coefficient multiplication is \(3.02\times5.9 = 17.818\) and the exponent addition is \(3 + 4=7\). So the calculator answer is \(17.818\times10^{7}\). Adjust to proper scientific notation: \(1.7818\times10^{8}\).

Step 2: Round to significant figures

The original numbers \(3.02\) (3 sig figs) and \(5.9\) (2 sig figs). When multiplying, the result should have 2 sig figs. So we round \(1.7818\times10^{8}\) to \(1.8\times10^{8}\).

Calculator Answer: \(1.7818\times10^{8}\)

Rounded Answer: \(1.8\times10^{8}\)

Part c

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

Step 1: Multiply the numbers

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

Step 2: Round to significant figures

\(56\) has 2 sig figs and \(19.3\) has 3 sig figs. The result should have 2 sig figs. So we round \(1080.8\) to \(1100\space m^{2}\) (or \(1.1\times10^{3}\space m^{2}\))

Calculator Answer: \(1080.8\space m^{2}\)

Rounded Answer: \(1.1\times10^{3}\space m^{2}\) (or \(1100\space m^{2}\))

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

Answer:

\(\approx4.83\%\) (or more precisely, using the formula: \(\frac{\vert8.25 - 7.87\vert}{7.87}\times100=\frac{0.38}{7.87}\times100\approx4.83\%\))

Part 6: Scientific Notation and Regular Notation Conversion

1. \(1.230\times10^{-8}\) to Regular Notation