{"id":2940,"date":"2024-08-27T10:15:39","date_gmt":"2024-08-27T10:15:39","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=2940"},"modified":"2024-11-20T00:52:56","modified_gmt":"2024-11-20T00:52:56","slug":"exponents-and-scientific-notation-learn-it-6","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/exponents-and-scientific-notation-learn-it-6\/","title":{"raw":"Exponents and Scientific Notation: Learn It 6","rendered":"Exponents and Scientific Notation: Learn It 6"},"content":{"raw":"

Multiplying and Dividing Numbers in Scientific Notation<\/h2>\r\nWhen multiplying and dividing numbers in scientific notation, we apply the basic principles of exponents, which streamline these operations and prevent errors that could arise from manual calculation with long numbers. Understanding how to correctly multiply and divide in scientific notation is crucial for maintaining precision and efficiency in scientific calculations.\r\n\r\n
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multiplying and dividing numbers in scientific notation<\/h3>\r\n
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  • To multiply numbers in scientific notation, we need to multiply the coefficients and add the powers of [latex]10[\/latex].<\/li>\r\n \t
  • To divide numbers in scientific notation, we need to divide the coefficients and subtract the powers of [latex]10[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>
    You might be wondering \"How are we are able to simply multiply the decimal terms and add the exponents? What properties of numbers enable this?\"Recall that multiplication is both commutative and associative - this means, as long as multiplication is the only operation being performed, we can move the factors around to suit our needs. Lastly, the product rule for exponents, [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex], allows us to add the exponents on the base of [latex]10[\/latex].\r\n\r\n<\/section>
    In April 2014, the population of the United States was about [latex]308,000,000[\/latex] people. The national debt was about [latex]$17,547,000,000,000[\/latex]. Write each number in scientific notation, rounding figures to two decimal places, and find the amount of the debt per U.S. citizen. Write the answer in both scientific and standard notations.[reveal-answer q=\"500341\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"500341\"]The population was [latex]308,000,000=3.08\\times {10}^{8}[\/latex].The national debt was [latex]\\$ 17,547,000,000,000 \\approx \\$1.75 \\times 10^{13}[\/latex].To find the amount of debt per citizen, divide the national debt by the number of citizens.\r\n
    [latex]\\begin{align} \\left(1.75\\times {10}^{13}\\right)\\div \\left(3.08\\times {10}^{8}\\right)& = \\left(\\frac{1.75}{3.08}\\right)\\cdot \\left(\\frac{{10}^{13}}{{10}^{8}}\\right) \\\\ & \\approx 0.57\\times {10}^{5}\\hfill \\\\ & = 5.7\\times {10}^{4} \\end{align}[\/latex]<\/div>\r\nThe debt per citizen at the time was about [latex]\\$5.7\\times {10}^{4}[\/latex], or [latex]$57,000[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
    [ohm2_question hide_question_numbers=1]1843[\/ohm2_question]<\/section>
    [ohm2_question hide_question_numbers=1]1844[\/ohm2_question]<\/section>","rendered":"

    Multiplying and Dividing Numbers in Scientific Notation<\/h2>\n

    When multiplying and dividing numbers in scientific notation, we apply the basic principles of exponents, which streamline these operations and prevent errors that could arise from manual calculation with long numbers. Understanding how to correctly multiply and divide in scientific notation is crucial for maintaining precision and efficiency in scientific calculations.<\/p>\n

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    multiplying and dividing numbers in scientific notation<\/h3>\n
      \n
    • To multiply numbers in scientific notation, we need to multiply the coefficients and add the powers of [latex]10[\/latex].<\/li>\n
    • To divide numbers in scientific notation, we need to divide the coefficients and subtract the powers of [latex]10[\/latex].<\/li>\n<\/ul>\n<\/div>\n<\/section>\n
      You might be wondering “How are we are able to simply multiply the decimal terms and add the exponents? What properties of numbers enable this?”Recall that multiplication is both commutative and associative – this means, as long as multiplication is the only operation being performed, we can move the factors around to suit our needs. Lastly, the product rule for exponents, [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex], allows us to add the exponents on the base of [latex]10[\/latex].<\/p>\n<\/section>\n
      In April 2014, the population of the United States was about [latex]308,000,000[\/latex] people. The national debt was about [latex]$17,547,000,000,000[\/latex]. Write each number in scientific notation, rounding figures to two decimal places, and find the amount of the debt per U.S. citizen. Write the answer in both scientific and standard notations.<\/p>\n