{"id":67,"date":"2024-10-16T18:27:45","date_gmt":"2024-10-16T18:27:45","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/?post_type=chapter&p=67"},"modified":"2024-10-16T18:27:46","modified_gmt":"2024-10-16T18:27:46","slug":"exponents-and-scientific-notation-learn-it-7","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/exponents-and-scientific-notation-learn-it-7\/","title":{"raw":"Exponents and Scientific Notation: Learn It 7","rendered":"Exponents and Scientific Notation: Learn It 7"},"content":{"raw":"

Using Scientific Notation in Applications<\/h2>\r\nNow that we've mastered converting between standard and scientific notation, let\u2019s explore how this skill can be applied in real-world scenarios.\r\n\r\n
[ohm2_question hide_question_numbers=1]18744[\/ohm2_question]<\/section>
Let's dive into some practical exercises to see scientific notation in action, helping us solve problems efficiently in science, engineering, and beyond.<\/section>
[ohm2_question hide_question_numbers=1]18739[\/ohm2_question]<\/section>
Suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen).The average drop of water contains around [latex]1.32\\times {10}^{21}[\/latex] molecules of water and 1 L of water holds about [latex]1.22\\times {10}^{4}[\/latex] average drops.Therefore, there are approximately [latex]\\left(3\\right)\\cdot\\left(1.32\\times {10}^{21}\\right)\\cdot \\left(1.22\\times {10}^{4}\\right)=\\left(3\\cdot1.32\\cdot1.22\\right)\\times\\left({10}^{4}\\cdot{10}^{25}\\right)\\approx 4.83\\times {10}^{25}[\/latex] atoms in 1 L of water.\r\n\r\nWe simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!\r\n\r\nNote: How are we are able to simply multiply the decimal terms and add the exponents? What properties of numbers enable this?\r\n

Recall that multiplication is both [pb_glossary id=\"615\"]commutative[\/pb_glossary] and [pb_glossary id=\"616\"]associative[\/pb_glossary]. That means, as long as multiplication is the only operation being performed, we can move the factors around to suit our needs. Lastly, the [pb_glossary id=\"617\"]product rule for exponents[\/pb_glossary] allows us to add the exponents on the base of [latex]10[\/latex].<\/p>\r\n\r\n<\/section>

Perform the operations and write the answer in scientific notation.\r\n
    \r\n \t
  1. [latex]\\left(8.14\\times {10}^{-7}\\right)\\left(6.5\\times {10}^{10}\\right)[\/latex]<\/li>\r\n \t
  2. [latex]\\left(4\\times {10}^{5}\\right)\\div \\left(-1.52\\times {10}^{9}\\right)[\/latex]<\/li>\r\n \t
  3. [latex]\\left(2.7\\times {10}^{5}\\right)\\left(6.04\\times {10}^{13}\\right)[\/latex]<\/li>\r\n \t
  4. [latex]\\left(1.2\\times {10}^{8}\\right)\\div \\left(9.6\\times {10}^{5}\\right)[\/latex]<\/li>\r\n \t
  5. [latex]\\left(3.33\\times {10}^{4}\\right)\\left(-1.05\\times {10}^{7}\\right)\\left(5.62\\times {10}^{5}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"380183\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"380183\"]\r\n
      \r\n \t
    1. [latex]\\begin{align}\\left(8.14 \\times 10^{-7}\\right)\\left(6.5 \\times 10^{10}\\right) & =\\left(8.14 \\times 6.5\\right)\\left(10^{-7} \\times 10^{10}\\right) && \\text{Commutative and associative properties of multiplication} \\\\ & =\\left(52.91\\right)\\left(10^{3}\\right) && \\text{Product rule of exponents} \\\\ & =5.291 \\times 10^{4} && \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/li>\r\n \t
    2. [latex]\\begin{align} \\left(4\\times {10}^{5}\\right)\\div \\left(-1.52\\times {10}^{9}\\right)& = \\left(\\frac{4}{-1.52}\\right)\\left(\\frac{{10}^{5}}{{10}^{9}}\\right)&& \\text{Commutative and associative properties of multiplication} \\\\ & = \\left(-2.63\\right)\\left({10}^{-4}\\right)&& \\text{Quotient rule of exponents} \\\\ & = -2.63\\times {10}^{-4}&& \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/li>\r\n \t
    3. [latex]\\begin{align} \\left(2.7\\times {10}^{5}\\right)\\left(6.04\\times {10}^{13}\\right)& = \\left(2.7\\times 6.04\\right)\\left({10}^{5}\\times {10}^{13}\\right)&& \\text{Commutative and associative properties of multiplication} \\\\ & = \\left(16.308\\right)\\left({10}^{18}\\right)&& \\text{Product rule of exponents} \\\\ & = 1.6308\\times {10}^{19}&& \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/li>\r\n \t
    4. [latex]\\begin{align} \\left(1.2\\times {10}^{8}\\right)\\div \\left(9.6\\times {10}^{5}\\right)& = \\left(\\frac{1.2}{9.6}\\right)\\left(\\frac{{10}^{8}}{{10}^{5}}\\right)&& \\text{Commutative and associative properties of multiplication} \\\\ & = \\left(0.125\\right)\\left({10}^{3}\\right)&& \\text{Quotient rule of exponents} \\\\ & = 1.25\\times {10}^{2}&& \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/li>\r\n \t
    5. [latex]\\begin{align} \\left(3.33\\times {10}^{4}\\right)\\left(-1.05\\times {10}^{7}\\right)\\left(5.62\\times {10}^{5}\\right)& = \\left[3.33\\times \\left(-1.05\\right)\\times 5.62\\right]\\left({10}^{4}\\times {10}^{7}\\times {10}^{5}\\right) \\\\ & \\approx \\left(-19.65\\right)\\left({10}^{16}\\right) \\\\ & = -1.965\\times {10}^{17} \\end{align}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
      [ohm2_question hide_question_numbers=1]18747[\/ohm2_question]<\/section>","rendered":"

      Using Scientific Notation in Applications<\/h2>\n

      Now that we’ve mastered converting between standard and scientific notation, let\u2019s explore how this skill can be applied in real-world scenarios.<\/p>\n