Scientific Notation, Metric System, & Unit Conversion Review ... - Free Printable

Problem Overview:

The worksheet focuses on scientific notation, the metric system, and unit conversion. The tasks involve rewriting numbers in scientific notation, performing calculations with scientific notation, and ensuring the correct use of significant figures.

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Task 1: Rewrite the following numbers in scientific notation

We need to rewrite each number in scientific notation, including units, and ensure the appropriate number of significant figures.

#### a. Altitude of summit of Mt. Kaʻala (highest point on Oʻahu): 4020 ft
- Step 1: Write the number in scientific notation.
\[
4020 = 4.02 \times 10^3
\]
- Step 2: Include the unit.
\[
4.02 \times 10^3 \, \text{ft}
\]

#### b. Altitude of summit of Mauna Kea: 13,796 ft
- Step 1: Write the number in scientific notation.
\[
13,796 = 1.3796 \times 10^4
\]
- Step 2: Include the unit.
\[
1.3796 \times 10^4 \, \text{ft}
\]

#### c. Thickness of a human hair: 0.015 cm
- Step 1: Write the number in scientific notation.
\[
0.015 = 1.5 \times 10^{-2}
\]
- Step 2: Include the unit.
\[
1.5 \times 10^{-2} \, \text{cm}
\]

#### d. Wavelength of reddish light: 0.0000007 m
- Step 1: Write the number in scientific notation.
\[
0.0000007 = 7 \times 10^{-7}
\]
- Step 2: Include the unit.
\[
7 \times 10^{-7} \, \text{m}
\]

#### e. Height of your instructor: 1.80 m
- Step 1: Write the number in scientific notation.
\[
1.80 = 1.80 \times 10^0
\]
- Step 2: Include the unit.
\[
1.80 \times 10^0 \, \text{m}
\]

#### f. Number of galaxies in the universe: 1 trillion galaxies
- Step 1: Convert "trillion" to numerical form.
\[
1 \, \text{trillion} = 1,000,000,000,000 = 1 \times 10^{12}
\]
- Step 2: Include the unit.
\[
1 \times 10^{12} \, \text{galaxies}
\]

#### g. Age of the universe in seconds: 430,000,000,000,000 s
- Step 1: Write the number in scientific notation.
\[
430,000,000,000,000 = 4.3 \times 10^{14}
\]
- Step 2: Include the unit.
\[
4.3 \times 10^{14} \, \text{s}
\]

#### h. Volume of a hydrogen atom: \(0.000\,000\,000\,000\,000\,000\,000\,062\,1 \, \text{cm}^3\)
- Step 1: Write the number in scientific notation.
\[
0.000\,000\,000\,000\,000\,000\,000\,062\,1 = 6.21 \times 10^{-24}
\]
- Step 2: Include the unit.
\[
6.21 \times 10^{-24} \, \text{cm}^3
\]

---

Task 2: Calculate the following and write your answer in scientific notation

We need to perform calculations involving addition, subtraction, multiplication, and division, and express the results in scientific notation with the correct number of significant figures.

#### a. \(10^{10} \times 10^4\)
- Step 1: Use the rule for multiplying powers of 10: \(10^a \times 10^b = 10^{a+b}\).
\[
10^{10} \times 10^4 = 10^{10+4} = 10^{14}
\]
- Answer:
\[
10^{14}
\]

#### b. \(10^{10} \div 10^4\)
- Step 1: Use the rule for dividing powers of 10: \(10^a \div 10^b = 10^{a-b}\).
\[
10^{10} \div 10^4 = 10^{10-4} = 10^6
\]
- Answer:
\[
10^6
\]

#### c. \(10^{10} + 10^4\)
- Step 1: Convert both terms to the same power of 10 for addition.
\[
10^{10} = 10,000,000,000
\]
\[
10^4 = 10,000
\]
\[
10^{10} + 10^4 = 10,000,000,000 + 10,000 = 10,000,010,000
\]
- Step 2: Write the result in scientific notation.
\[
10,000,010,000 = 1.000001 \times 10^{10}
\]
- Answer:
\[
1.000001 \times 10^{10}
\]

#### d. \(10^{10} - 10^4\)
- Step 1: Convert both terms to the same power of 10 for subtraction.
\[
10^{10} = 10,000,000,000
\]
\[
10^4 = 10,000
\]
\[
10^{10} - 10^4 = 10,000,000,000 - 10,000 = 9,999,990,000
\]
- Step 2: Write the result in scientific notation.
\[
9,999,990,000 = 9.99999 \times 10^9
\]
- Answer:
\[
9.99999 \times 10^9
\]

#### e. \(10^{10} - 10^4\)
- This is the same as part (d). The result is:
\[
9.99999 \times 10^9
\]

#### f. \((2 \times 10^5) \times (3 \times 10^7)\)
- Step 1: Multiply the coefficients and add the exponents.
\[
(2 \times 10^5) \times (3 \times 10^7) = (2 \times 3) \times (10^5 \times 10^7) = 6 \times 10^{5+7} = 6 \times 10^{12}
\]
- Answer:
\[
6 \times 10^{12}
\]

#### g. \((3.5 \times 10^{17}) \div (7 \times 10^8)\)
- Step 1: Divide the coefficients and subtract the exponents.
\[
(3.5 \times 10^{17}) \div (7 \times 10^8) = \left(\frac{3.5}{7}\right) \times \left(10^{17} \div 10^8\right) = 0.5 \times 10^{17-8} = 0.5 \times 10^9
\]
- Step 2: Write the result in proper scientific notation.
\[
0.5 \times 10^9 = 5 \times 10^8
\]
- Answer:
\[
5 \times 10^8
\]

#### h. \(10^7 - (3 \times 10^9)\)
- Step 1: Convert both terms to the same power of 10 for subtraction.
\[
10^7 = 10,000,000
\]
\[
3 \times 10^9 = 3,000,000,000
\]
\[
10^7 - (3 \times 10^9) = 10,000,000 - 3,000,000,000 = -2,990,000,000
\]
- Step 2: Write the result in scientific notation.
\[
-2,990,000,000 = -2.99 \times 10^9
\]
- Answer:
\[
-2.99 \times 10^9
\]

#### i. \((42.3 \times 10^{-9}) + (5.77 \times 10^{-6})\)
- Step 1: Convert both terms to the same power of 10 for addition.
\[
42.3 \times 10^{-9} = 4.23 \times 10^{-8}
\]
\[
5.77 \times 10^{-6} = 577 \times 10^{-8}
\]
\[
(4.23 \times 10^{-8}) + (577 \times 10^{-8}) = (4.23 + 577) \times 10^{-8} = 581.23 \times 10^{-8}
\]
- Step 2: Write the result in proper scientific notation.
\[
581.23 \times 10^{-8} = 5.8123 \times 10^{-6}
\]
- Answer:
\[
5.8123 \times 10^{-6}
\]

#### j. \((34.9 \times 10^4) \times (2.12 \times 10^{-15})\)
- Step 1: Multiply the coefficients and add the exponents.
\[
(34.9 \times 10^4) \times (2.12 \times 10^{-15}) = (34.9 \times 2.12) \times (10^4 \times 10^{-15}) = 73.988 \times 10^{4-15} = 73.988 \times 10^{-11}
\]
- Step 2: Write the result in proper scientific notation.
\[
73.988 \times 10^{-11} = 7.3988 \times 10^{-10}
\]
- Answer:
\[
7.3988 \times 10^{-10}
\]

#### k. \((0.88 \times 10^{-5}) \times (6.3 \times 10^{-10})\)
- Step 1: Multiply the coefficients and add the exponents.
\[
(0.88 \times 10^{-5}) \times (6.3 \times 10^{-10}) = (0.88 \times 6.3) \times (10^{-5} \times 10^{-10}) = 5.544 \times 10^{-5-10} = 5.544 \times 10^{-15}
\]
- Answer:
\[
5.544 \times 10^{-15}
\]

#### l. \((9.876 \times 10^{29}) - (5.4321 \times 10^{-15})\)
- Step 1: Note that the second term is extremely small compared to the first term. Subtracting it has no significant effect on the larger term.
\[
(9.876 \times 10^{29}) - (5.4321 \times 10^{-15}) \approx 9.876 \times 10^{29}
\]
- Answer:
\[
9.876 \times 10^{29}
\]

#### m. Mass of Earth - mass of Moon: \((5.974 \times 10^{27} \, \text{g}) - (7.348 \times 10^{25} \, \text{g})\)
- Step 1: Convert both terms to the same power of 10 for subtraction.
\[
5.974 \times 10^{27} = 597.4 \times 10^{25}
\]
\[
7.348 \times 10^{25} = 7.348 \times 10^{25}
\]
\[
(597.4 \times 10^{25}) - (7.348 \times 10^{25}) = (597.4 - 7.348) \times 10^{25} = 589.052 \times 10^{25}
\]
- Step 2: Write the result in proper scientific notation.
\[
589.052 \times 10^{25} = 5.89052 \times 10^{27}
\]
- Answer:
\[
5.89052 \times 10^{27} \, \text{g}
\]

#### n. Mass of Earth - mass of Moon: \((5.974 \times 10^{27} \, \text{g}) - (7.348 \times 10^{25} \, \text{g})\)
- This is the same as part (m). The result is:
\[
5.89052 \times 10^{27} \, \text{g}
\]

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Final Answers:

1.
- a. \(4.02 \times 10^3 \, \text{ft}\)
- b. \(1.3796 \times 10^4 \, \text{ft}\)
- c. \(1.5 \times 10^{-2} \, \text{cm}\)
- d. \(7 \times 10^{-7} \, \text{m}\)
- e. \(1.80 \times 10^0 \, \text{m}\)
- f. \(1 \times 10^{12} \, \text{galaxies}\)
- g. \(4.3 \times 10^{14} \, \text{s}\)
- h. \(6.21 \times 10^{-24} \, \text{cm}^3\)

2.
- a. \(10^{14}\)
- b. \(10^6\)
- c. \(1.000001 \times 10^{10}\)
- d. \(9.99999 \times 10^9\)
- e. \(9.99999 \times 10^9\)
- f. \(6 \times 10^{12}\)
- g. \(5 \times 10^8\)
- h. \(-2.99 \times 10^9\)
- i. \(5.8123 \times 10^{-6}\)
- j. \(7.3988 \times 10^{-10}\)
- k. \(5.544 \times 10^{-15}\)
- l. \(9.876 \times 10^{29}\)
- m. \(5.89052 \times 10^{27} \, \text{g}\)
- n. \(5.89052 \times 10^{27} \, \text{g}\)

\boxed{\text{See detailed solutions above}}