{"id":593,"date":"2024-04-23T17:26:57","date_gmt":"2024-04-23T17:26:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=593"},"modified":"2025-08-21T23:01:02","modified_gmt":"2025-08-21T23:01:02","slug":"exponents-and-scientific-notation-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/exponents-and-scientific-notation-learn-it-5\/","title":{"raw":"Exponents and Scientific Notation: Learn It 5","rendered":"Exponents and Scientific Notation: Learn It 5"},"content":{"raw":"

Scientific Notation<\/h2>\r\nScientific notation<\/strong> is a way of writing numbers that are very large or very small in a compact and convenient form. It is widely used in the sciences, engineering, and mathematics, and is especially useful for dealing with numbers that have a lot of zeros.\r\n\r\n
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scientific notation<\/h3>\r\nIn scientific notation, a number is written as a product of two parts: a coefficient and a power of [latex]10[\/latex].\r\n\r\n \r\n\r\nIt is written in the form [latex]a\\times {10}^{n}[\/latex], where [latex]1\\le |a|<10[\/latex] and [latex]n[\/latex] is an integer.\r\n\r\n \r\n\r\n[latex]n[\/latex] can be positive or negative depending on whether a small or large number is being converted to scientific notation.\r\n\r\n<\/div>\r\n<\/section>
When converting large numbers, [latex]n[\/latex] is positive. When converting small numbers, [latex]n[\/latex] is negative.<\/section>
Consider the number [latex]2,780,418[\/latex], which is currently written in standard notation. How do you think it is expressed in scientific notation?\r\n[reveal-answer q=\"500341\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"500341\"]\r\nThe number [latex]2,780,418[\/latex] in standard notation can be written in scientific notation as [latex]2.780418 x 10^6[\/latex], where [latex]2.780418[\/latex] is the coefficient and [latex]10^6[\/latex] represents [latex]10[\/latex] multiplied by itself [latex]6[\/latex] times [latex](1,000,000)[\/latex]This was obtained by moving the decimal left until it is to the right of the first nonzero digit, which is [latex]2[\/latex].
\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"The 2,780,418 becomes 2.780418 when the decimal shifts 6 spots[\/caption]\r\n\r\n<\/center>We obtain [latex]2.780418[\/latex] by moving the decimal point [latex]6[\/latex] places to the left. [latex]6[\/latex] is positive because we moved the decimal point to the left. This is what we should expect for a large number.[\/hidden-answer]<\/section>
How to: Convert Numbers from Standard Notation to Scientific Notation<\/strong>Although scientific notation can look confusing, we can follow a few simple steps to convert numbers in standard notation to scientific notation.\r\n
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  1. Count the number of places that the decimal point needs to be moved to the right or left so that the coefficient is between [latex]1[\/latex] and [latex]9[\/latex].<\/li>\r\n \t
  2. Move the decimal point the appropriate number of places to the right or left, and adjust the power of [latex]10[\/latex] accordingly.<\/li>\r\n \t
  3. Write the result as the coefficient followed by [latex]x 10^n[\/latex], where [latex]n[\/latex] represents the number of times that the decimal point was moved.<\/li>\r\n<\/ol>\r\nRemember the sign on [latex]n[\/latex] depends on if you are converting a small or large number to scientific notation. A rule to remember is if you move your decimal to the left [latex]n[\/latex] is positive and if you move your decimal to the right [latex]n[\/latex] is negative.\r\n\r\n<\/section>
    [ohm2_question hide_question_numbers=1]1838[\/ohm2_question]<\/section>
    How to: Convert Numbers from Scientific Notation to Standard Notation<\/strong>To convert from scientific notation to standard notation, we simply need to do the reverse of the steps given before.\r\n
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    1. Write the coefficient and the power of [latex]10[\/latex] separately.<\/li>\r\n \t
    2. Move the decimal point in the coefficient the same number of places to the left or right as the power of [latex]10[\/latex] suggests.<\/li>\r\n \t
    3. Combine the coefficient and the decimal point to get the standard notation representation of the number.<\/li>\r\n<\/ol>\r\n<\/section>
      [videopicker divId=\"tnh-video-picker\" title=\"Scientific Notation\" label=\"Select Video\"]\r\n[videooption displayName=\"How to convert a number to scientific notation\" value=\"\/\/plugin.3playmedia.com\/show?mf=12425234&p3sdk_version=1.10.1&p=20361&pt=375&video_id=J-eQb7NMJFA&video_target=tpm-plugin-g1pfjtnh-J-eQb7NMJFA\"][videooption displayName=\"Converting to Scientific Notation Examples!\" value=\"\/\/plugin.3playmedia.com\/show?mf=12425235;p3sdk_version=1.10.1&p=20361&pt=375&video_id=Wf-HlVqZPHY&video_target=tpm-plugin-b4464ab5-Wf-HlVqZPHY\"] [videooption displayName=\"Math Antics - Scientific Notation\" value=\"\/\/plugin.3playmedia.com\/show?mf=12843106;p3sdk_version=1.10.1&p=20361&pt=375&video_id=bXkewQ7WEdI&video_target=tpm-plugin-vhx08cmh-bXkewQ7WEdI\"]\r\n[\/videopicker]You can view the transcript for \u201cHow to convert a number to scientific notation\u201d here (opens in new window).<\/a>\r\nYou can view the transcript for \u201cConverting to Scientific Notation Examples!\u201d here (opens in new window).<\/a>\r\nYou can view the transcript for \u201cMath Antics - Scientific Notation\u201d here (opens in new window).<\/a><\/section>
      [ohm2_question hide_question_numbers=1]1840[\/ohm2_question]<\/section>
      [ohm2_question hide_question_numbers=1]1841[\/ohm2_question]<\/section>","rendered":"

      Scientific Notation<\/h2>\n

      Scientific notation<\/strong> is a way of writing numbers that are very large or very small in a compact and convenient form. It is widely used in the sciences, engineering, and mathematics, and is especially useful for dealing with numbers that have a lot of zeros.<\/p>\n

      \n
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      scientific notation<\/h3>\n

      In scientific notation, a number is written as a product of two parts: a coefficient and a power of [latex]10[\/latex].<\/p>\n

       <\/p>\n

      It is written in the form [latex]a\\times {10}^{n}[\/latex], where [latex]1\\le |a|<10[\/latex] and [latex]n[\/latex] is an integer.\n\n \n\n[latex]n[\/latex] can be positive or negative depending on whether a small or large number is being converted to scientific notation.\n\n<\/div>\n<\/section>\n

      When converting large numbers, [latex]n[\/latex] is positive. When converting small numbers, [latex]n[\/latex] is negative.<\/section>\n
      Consider the number [latex]2,780,418[\/latex], which is currently written in standard notation. How do you think it is expressed in scientific notation?<\/p>\n