Intermediate Algebra Problems with Scientific Notation
Real-World Examples - Sample 5

Problem: Convert the following numbers from decimal notation to scientific notation:

1. \( 234 \)
2. \( 0.000052 \)
3. \( -220{,}000 \)
4. \( -0.00000000000000182 \)

Solutions:

1. \( 234 = 2.34 \times 10^2 \)
2. \( 0.000052 = 5.2 \times 10^{-5} \)
3. \( -220{,}000 = -2.2 \times 10^5 \)
4. \( -0.00000000000000182 = -1.82 \times 10^{-15} \)

Problem: Convert the following numbers from scientific notation to decimal notation:

1. \( 1.01 \times 10^2 \)
2. \( 8.2 \times 10^{-6} \)
3. \( 3.456 \times 10^{10} \)
4. \( -1.2 \times 10^{-8} \)

Solutions:

1. \( 1.01 \times 10^2 = 101 \)
2. \( 8.2 \times 10^{-6} = 0.0000082 \)
3. \( 3.456 \times 10^{10} = 34{,}560{,}000{,}000 \)
4. \( -1.2 \times 10^{-8} = -0.000000012 \)

Problem: Perform the multiplication and write the result in scientific notation:

1. \( (5 \times 10^2)(7 \times 10^5) \)
2. \( (1.2 \times 10^{-6})(3.5 \times 10^{14}) \)
3. \( (-4.5 \times 10^{-7})(8.62 \times 10^{10}) \)
4. \( (-7.2 \times 10^{-6})(0.00039) \)

Solutions:

1. \( (5 \times 10^2)(7 \times 10^5) = (5 \times 7)(10^2 \times 10^5) = 35 \times 10^7 = 3.5 \times 10^8 \)
2. \( (1.2 \times 10^{-6})(3.5 \times 10^{14}) = (1.2 \times 3.5)(10^{-6} \times 10^{14}) = 4.2 \times 10^8 \)
3. \( (-4.5 \times 10^{-7})(8.62 \times 10^{10}) = (-4.5 \times 8.62)(10^{-7} \times 10^{10}) = -38.79 \times 10^3 = -3.879 \times 10^4 \)
4. \( (-7.2 \times 10^{-6})(0.00039) = (-7.2 \times 10^{-6})(3.9 \times 10^{-4}) = (-7.2 \times 3.9)(10^{-6} \times 10^{-4}) = -28.08 \times 10^{-10} = -2.808 \times 10^{-9} \)

Problem: Perform the division and write the result in scientific notation:

1. \( \dfrac{4 \times 10^{13}}{5 \times 10^{-4}} \)
2. \( \dfrac{0.000012}{2.4 \times 10^{-8}} \)
3. \( \dfrac{-1.2 \times 10^{-6}}{1.5 \times 10^{-10}} \)
4. \( \dfrac{0.00000065}{0.000013} \)

Solutions:

1. \( \dfrac{4 \times 10^{13}}{5 \times 10^{-4}} = \dfrac{4}{5} \times \dfrac{10^{13}}{10^{-4}} = 0.8 \times 10^{17} = 8.0 \times 10^{16} \)
2. \( \dfrac{0.000012}{2.4 \times 10^{-8}} = \dfrac{1.2 \times 10^{-5}}{2.4 \times 10^{-8}} = 0.5 \times 10^3 = 5.0 \times 10^2 \)
3. \( \dfrac{-1.2 \times 10^{-6}}{1.5 \times 10^{-10}} = \dfrac{-1.2}{1.5} \times \dfrac{10^{-6}}{10^{-10}} = -0.8 \times 10^4 = -8.0 \times 10^3 \)
4. \( \dfrac{0.00000065}{0.000013} = \dfrac{6.5 \times 10^{-7}}{1.3 \times 10^{-5}} = 0.05 = 5.0 \times 10^{-2} \)

Problem: Evaluate and express the result in scientific notation:

1. \( (1.3 \times 10^{-6})^2 \)
2. \( (1.5 \times 10^{-8})^2(2.1 \times 10^5) \)
3. \( \dfrac{(1.144 \times 10^2)(6 \times 10^7)}{2.2 \times 10^4} \)
4. \( \dfrac{(0.00001)(2{,}500)}{(0.0003)(150{,}000)} \)

Solutions:

1. \( (1.3 \times 10^{-6})^2 = 1.3^2 \times (10^{-6})^2 = 1.69 \times 10^{-12} \)
2. \( (1.5 \times 10^{-8})^2(2.1 \times 10^5) = (2.25 \times 10^{-16})(2.1 \times 10^5) = 4.725 \times 10^{-11} \)
3. \[ \dfrac{(1.144 \times 10^2)(6 \times 10^7)}{2.2 \times 10^4} = \dfrac{6.864 \times 10^9}{2.2 \times 10^4} = 3.12 \times 10^5 \]
4. \[ \dfrac{(0.00001)(2500)}{(0.0003)(150000)} = \dfrac{(1.0 \times 10^{-5})(2.5 \times 10^3)}{(3.0 \times 10^{-4})(1.5 \times 10^5)} = \dfrac{2.5 \times 10^{-2}}{4.5 \times 10^1} \approx 0.556 \times 10^{-3} = 5.56 \times 10^{-4} \]

Problem: Evaluate the following real-world values and calculations in scientific notation:

1. Diameter of an atom: \( 0.0000000000001 \) meter
2. Diameter of Earth: \( 12{,}756{,}000 \) meters
3. 1 billion: \( 1{,}000{,}000{,}000 \)
4. 1 micron: \( 0.000001 \) meter
5. 1 trillion: \( 1{,}000{,}000{,}000{,}000 \)
6. Ratio: \( \dfrac{\text{Diameter of Earth}}{\text{Diameter of atom}} \)
7. Speed of light: \( 300{,}000{,}000 \) m/sec
8. Distance traveled by light in one year

Solutions:

1. Diameter of an atom: \( 1.0 \times 10^{-13} \text{ meters} \)
2. Diameter of Earth: \( 1.2756 \times 10^7 \text{ meters} \)
3. 1 billion: \( 1.0 \times 10^9 \)
4. 1 micron: \( 1.0 \times 10^{-6} \text{ meters} \)
5. 1 trillion: \( 1.0 \times 10^{12} \)
6. Ratio: \[ \dfrac{1.2756 \times 10^7}{1.0 \times 10^{-13}} = 1.2756 \times 10^{20} \]
7. Speed of light: \( 3.0 \times 10^8 \text{ m/sec} \)
8. Distance traveled by light in one year:
   Time in one year: \( 365 \times 24 \times 60 \times 60 = 3.1536 \times 10^7 \text{ seconds} \)
   Distance = \(\text{Time} \times \text{Speed} = (3.1536 \times 10^7)(3.0 \times 10^8) = 9.4608 \times 10^{15} \text{ meters} = 9.4608 \times 10^{12} \text{ kilometers} \)