{"id":9897,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9897"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"exponents-and-small-number","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/exponents-and-small-number\/","title":{"rendered":"Exponents And Small Number"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Exponents & Powers<\/strong><\/h2>\n

Chapter: <\/strong>Exponents & Small Numbers<\/strong><\/h3>\n

Reference: – What are Small Numbers, Negative Exponents for Small Numbers, Standard Form for Small Numbers (Scientific Notation), Converting Decimal to Standard Form, Converting Standard Form to Decimal, Comparing Very Small Numbers, Ordering Small Numbers, Real-Life Examples (Microscopic Sizes), Solved Examples, Odd-One-Out Problems, Common Mistakes<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • How to Write Very Small Numbers Using Negative Exponents<\/em><\/li>\n
  • How to Convert Small Decimals to Standard Form (A \u00d7 10<\/em>\u207b<\/em>\u207f<\/em>)<\/em><\/li>\n
  • How to Convert Standard Form with Negative Exponents to Decimals<\/em><\/li>\n
  • How to Compare and Order Very Small Numbers<\/em><\/li>\n<\/ul>\n

    Introduction to Exponents & Small Numbers<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    Small numbers are numbers between 0 and 1 (like 0.001, 0.00005, 0.0000002). Using negative exponents, we can write these small numbers in a compact form called standard form (scientific notation). For example, 0.001 = 1 \u00d7 10\u207b\u00b3, where the negative exponent tells us how many places the decimal point moved to the right.<\/p>\n

    When we write small numbers using exponents, we essentially ask:<\/p>\n

    “How can I express this tiny number in a shorter, easier-to-read way?”<\/p>\n

    Negative exponents are the key to representing very small numbers efficiently.<\/p>\n

    Importance of Writing Small Numbers with Exponents<\/u><\/strong><\/p>\n

      \n
    • Used in science (size of bacteria, viruses, atoms)<\/li>\n
    • Used in medicine (doses of medicine, cell sizes)<\/li>\n
    • Used in physics (wavelengths of light, atomic particles)<\/li>\n
    • Used in chemistry (molecular sizes, concentrations)<\/li>\n
    • Makes calculations with very small numbers easier<\/li>\n<\/ul>\n

      Example<\/strong><\/p>\n

      The size of a red blood cell is about 0.000007 m. In standard form: 7 \u00d7 10\u207b\u2076 m.
      \nThe mass of a dust particle is about 0.0000000007 kg. In standard form: 7 \u00d7 10\u207b\u00b9\u2070 kg.<\/p>\n

      Subtopics<\/u><\/strong><\/p>\n

      1. Understanding Negative Exponents for Small Numbers<\/strong><\/p>\n

      A negative exponent tells us that the number is less than 1.<\/p>\n

      Pattern:<\/strong><\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Positive Exponent<\/p>\n<\/td>\n

      		\n

      Value<\/p>\n<\/td>\n

      		\n

      Negative Exponent<\/p>\n<\/td>\n

      		\n

      Value<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      10\u00b9 = 10<\/p>\n<\/td>\n

      		\n

      10<\/p>\n<\/td>\n

      		\n

      10\u207b\u00b9 = 1\/10<\/p>\n<\/td>\n

      		\n

      0.1<\/p>\n<\/td>\n<\/tr>\n

      \n

      10\u00b2 = 100<\/p>\n<\/td>\n

      		\n

      100<\/p>\n<\/td>\n

      		\n

      10\u207b\u00b2 = 1\/100<\/p>\n<\/td>\n

      		\n

      0.01<\/p>\n<\/td>\n<\/tr>\n

      \n

      10\u00b3 = 1000<\/p>\n<\/td>\n

      		\n

      1000<\/p>\n<\/td>\n

      		\n

      10\u207b\u00b3 = 1\/1000<\/p>\n<\/td>\n

      		\n

      0.001<\/p>\n<\/td>\n<\/tr>\n

      \n

      10\u2074 = 10000<\/p>\n<\/td>\n

      		\n

      10000<\/p>\n<\/td>\n

      		\n

      10\u207b\u2074 = 1\/10000<\/p>\n<\/td>\n

      		\n

      0.0001<\/p>\n<\/td>\n<\/tr>\n

      \n

      10\u2075 = 100000<\/p>\n<\/td>\n

      		\n

      100000<\/p>\n<\/td>\n

      		\n

      10\u207b\u2075 = 1\/100000<\/p>\n<\/td>\n

      		\n

      0.00001<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Key Observation:<\/strong>\u00a0The negative exponent tells the number of decimal places after the decimal point before the first non-zero digit.<\/p>\n

      Example:<\/strong>\u00a010\u207b\u00b3 = 0.001 (3 decimal places before the 1)<\/p>\n

      2. Standard Form for Small Numbers (Scientific Notation)<\/strong><\/p>\n

      A very small number is written in standard form as:\u00a0A \u00d7 10^(-n)<\/strong>\u00a0where:<\/p>\n

        \n
      • 1 \u2264 A < 10 (A is a number between 1 and 10, can be a decimal)<\/li>\n
      • n is a positive integer (the number of places the decimal moved)<\/li>\n<\/ul>\n

        Rules for Writing Small Numbers in Standard Form:<\/strong><\/p>\n

        Step 1:\u00a0Move the decimal point to the right until you have a number between 1 and 10.<\/p>\n

        Step 2:\u00a0Count how many places you moved the decimal point. That number becomes n.<\/p>\n

        Step 3:\u00a0Write the number as A \u00d7 10^(-n).<\/p>\n

        Example 1:\u00a0Write 0.005 in standard form<\/p>\n

        0.005 \u2192 move decimal 3 places right \u2192 5 \u2192 between 1 and 10
        \n0.005 = 5 \u00d7 10\u207b\u00b3<\/p>\n

        Example 2:\u00a0Write 0.00042 in standard form<\/p>\n

        0.00042 \u2192 move decimal 4 places right \u2192 4.2 \u2192 between 1 and 10
        \n0.00042 = 4.2 \u00d7 10\u207b\u2074<\/p>\n

        Example 3:\u00a0Write 0.0000003 in standard form<\/p>\n

        0.0000003 \u2192 move decimal 7 places right \u2192 3 \u2192 between 1 and 10
        \n0.0000003 = 3 \u00d7 10\u207b\u2077<\/p>\n

        Example 4:\u00a0Write 0.0000105 in standard form<\/p>\n

        0.0000105 \u2192 move decimal 5 places right \u2192 1.05 \u2192 between 1 and 10
        \n0.0000105 = 1.05 \u00d7 10\u207b\u2075<\/p>\n

        3. Converting Standard Form (Small Numbers) to Decimal<\/strong><\/p>\n

        To convert A \u00d7 10^(-n) to decimal form, move the decimal point n places to the left (add zeros as needed).<\/p>\n

        Example 1:\u00a03 \u00d7 10\u207b\u2074 = 0.0003 (move decimal 4 places left from 3.0)<\/p>\n

        Example 2:\u00a02.5 \u00d7 10\u207b\u00b3 = 0.0025<\/p>\n

        Example 3:\u00a01.23 \u00d7 10\u207b\u2075 = 0.0000123<\/p>\n

        Example 4:\u00a09 \u00d7 10\u207b\u2076 = 0.000009<\/p>\n

        4. Comparing Very Small Numbers<\/strong><\/p>\n

        When comparing numbers in standard form with negative exponents, remember: More negative exponent means smaller number.<\/p>\n

        Rule:<\/strong>\u00a0Compare the exponents first. The number with the larger (less negative) exponent is larger.<\/p>\n

        Order from largest to smallest:<\/strong>\u00a010\u207b\u00b2 > 10\u207b\u00b3 > 10\u207b\u2074<\/p>\n

        Examples:<\/strong><\/p>\n

        \n\n\n\n\n\n\n\n
        \n

        Number<\/p>\n<\/td>\n

        		\n

        Standard Form<\/p>\n<\/td>\n

        		\n

        Exponent<\/p>\n<\/td>\n

        		\n

        Size Order<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

        \n

        0.001<\/p>\n<\/td>\n

        		\n

        1 \u00d7 10\u207b\u00b3<\/p>\n<\/td>\n

        		\n

        -3<\/p>\n<\/td>\n

        		\n

        Largest (among these)<\/p>\n<\/td>\n<\/tr>\n

        \n

        0.0001<\/p>\n<\/td>\n

        		\n

        1 \u00d7 10\u207b\u2074<\/p>\n<\/td>\n

        		\n

        -4<\/p>\n<\/td>\n

        		\n

        Middle<\/p>\n<\/td>\n<\/tr>\n

        \n

        0.00001<\/p>\n<\/td>\n

        		\n

        1 \u00d7 10\u207b\u2075<\/p>\n<\/td>\n

        		\n

        -5<\/p>\n<\/td>\n

        		\n

        Smallest<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

        Source: Kapdec.com<\/p>\n<\/div>\n

        Comparing with different A values (same exponent):<\/strong>\u00a0Compare A values.<\/p>\n

        Example:<\/strong>\u00a0Which is larger: 3 \u00d7 10\u207b\u2074 or 5 \u00d7 10\u207b\u2074?
        \nExponents are same (-4). Compare 3 and 5 \u2192 5 \u00d7 10\u207b\u2074 is larger.<\/p>\n

        Comparing with different exponents:<\/strong>\u00a0Larger exponent (less negative) = larger number.<\/p>\n

        Example:<\/strong>\u00a0Which is larger: 2 \u00d7 10\u207b\u00b3 or 8 \u00d7 10\u207b\u2075?<\/p>\n

        2 \u00d7 10\u207b\u00b3 = 0.002, 8 \u00d7 10\u207b\u2075 = 0.00008 \u2192 2 \u00d7 10\u207b\u00b3 is larger because exponent -3 > -5<\/p>\n

        5. Ordering Small Numbers from Smallest to Largest<\/strong><\/p>\n

        To order small numbers, write them all in standard form with the same exponent, then compare.<\/p>\n

        Example:<\/strong>\u00a0Order 0.0005, 0.00003, 0.0002 from smallest to largest.<\/p>\n

        Write all in standard form: 5 \u00d7 10\u207b\u2074, 3 \u00d7 10\u207b\u2075, 2 \u00d7 10\u207b\u2074<\/p>\n

        Convert to same exponent (say 10\u207b\u2075): 50 \u00d7 10\u207b\u2075, 3 \u00d7 10\u207b\u2075, 20 \u00d7 10\u207b\u2075<\/p>\n

        Now order A values: 3, 20, 50 \u2192 3 \u00d7 10\u207b\u2075, 20 \u00d7 10\u207b\u2075, 50 \u00d7 10\u207b\u2075<\/p>\n

        So smallest to largest: 0.00003, 0.0002, 0.0005<\/p>\n

        Solved Examples<\/u><\/strong><\/p>\n

        Example 1 \u2013 Write in Standard Form:<\/strong>\u00a0Write 0.00007 in standard form.<\/p>\n

        Solution:<\/strong>\u00a0Move decimal 5 places right \u2192 7 \u2192 7 \u00d7 10\u207b\u2075<\/p>\n

        Answer:<\/strong>\u00a07 \u00d7 10\u207b\u2075<\/p>\n

        \u00a0<\/p>\n

        Example 2 \u2013 Write in Standard Form:<\/strong>\u00a0Write 0.0000234 in standard form.<\/p>\n

        Solution:<\/strong>\u00a0Move decimal 5 places right \u2192 2.34 \u2192 2.34 \u00d7 10\u207b\u2075<\/p>\n

        Answer:<\/strong>\u00a02.34 \u00d7 10\u207b\u2075<\/p>\n

        \u00a0<\/p>\n

        Example 3 \u2013 Convert to Decimal:<\/strong>\u00a0Write 4.5 \u00d7 10\u207b\u2076 as a decimal.<\/p>\n

        Solution:<\/strong>\u00a0Move decimal 6 places left: 0.0000045<\/p>\n

        Answer:<\/strong>\u00a00.0000045<\/p>\n

        \u00a0<\/p>\n

        Example 4 \u2013 Convert to Decimal:<\/strong>\u00a0Write 1.23 \u00d7 10\u207b\u2074 as a decimal.<\/p>\n

        Solution:<\/strong>\u00a0Move decimal 4 places left: 0.000123<\/p>\n

        Answer:<\/strong>\u00a00.000123<\/p>\n

        \u00a0<\/p>\n

        Example 5 \u2013 Compare:<\/strong>\u00a0Which is larger: 6 \u00d7 10\u207b\u2075 or 2 \u00d7 10\u207b\u2074?<\/p>\n

        Solution:<\/strong>\u00a0Exponents: -5 and -4. Since -4 > -5, 2 \u00d7 10\u207b\u2074 is larger.<\/p>\n

        Answer:<\/strong>\u00a02 \u00d7 10\u207b\u2074<\/p>\n

        \u00a0<\/p>\n

        Example 6 \u2013 Order:<\/strong>\u00a0Order from smallest to largest: 3 \u00d7 10\u207b\u2074, 8 \u00d7 10\u207b\u2076, 5 \u00d7 10\u207b\u2075<\/p>\n

        Solution:<\/strong>\u00a0Write all with exponent 10\u207b\u2076: 300 \u00d7 10\u207b\u2076, 8 \u00d7 10\u207b\u2076, 50 \u00d7 10\u207b\u2076
        \nOrder A values: 8, 50, 300
        \nSmallest to largest: 8 \u00d7 10\u207b\u2076, 5 \u00d7 10\u207b\u2075, 3 \u00d7 10\u207b\u2074<\/p>\n

        Answer:<\/strong>\u00a08 \u00d7 10\u207b\u2076, 5 \u00d7 10\u207b\u2075, 3 \u00d7 10\u207b\u2074<\/p>\n

        Common Mistakes to Avoid<\/u><\/strong><\/p>\n

        Mistake 1 \u2013 Moving decimal, the wrong way<\/strong>
        \n0.0005 = 5 \u00d7 10\u207b\u2074 (move RIGHT 4 places), NOT 5 \u00d7 10\u2074.
        \nCorrect understanding: Small numbers (between 0 and 1) use negative exponents.<\/p>\n

        Mistake 2 \u2013 Counting decimal places incorrectly<\/strong>
        \n0.0003 has 4 decimal places before 3? Actually 0.0003 = 3 \u00d7 10\u207b\u2074 (3 is in the 4th decimal place).
        \nCorrect understanding: Count how many places you move the decimal to get between 1 and 10.<\/p>\n

        Mistake 3 \u2013 Forgetting that A must be between 1 and 10<\/strong>
        \nWriting 0.5 \u00d7 10\u207b\u00b3 is incorrect standard form.
        \nCorrect understanding: 0.5 \u00d7 10\u207b\u00b3 = 5 \u00d7 10\u207b\u2074 (adjust exponent).<\/p>\n

        Mistake 4 \u2013 Confusing 10<\/strong>\u207b<\/strong>\u2074<\/strong> with 10<\/strong>\u2074<\/strong>
        \n10\u207b\u2074 = 0.0001, 10\u2074 = 10,000 (very different!).
        \nCorrect understanding: Negative exponent = number less than 1.<\/p>\n

        Mistake 5 \u2013 Comparing negative exponents incorrectly<\/strong>
        \nThinking 10\u207b\u2075 > 10\u207b\u2074 because 5 > 4. Wrong! -5 is less than -4.
        \nCorrect understanding: 10\u207b\u2074 = 0.0001, 10\u207b\u2075 = 0.00001, so 10\u207b\u2074 > 10\u207b\u2075.<\/p>\n

        Mistake 6 \u2013 Adding zeros when converting to decimal<\/strong>
        \n3 \u00d7 10\u207b\u2075 = 0.00003 (have 4 zeros before the 3? No, 5 decimal places total: 0.00003).
        \nCorrect understanding: Move decimal 5 places left from 3.0 \u2192 0.00003.<\/p>\n

        \u00a0<\/p>\n

        Quick Reference Summary<\/u><\/strong><\/p>\n

        Small Numbers (0 to 1):<\/strong>\u00a0Written as A \u00d7 10^(-n) where 1 \u2264 A < 10, n positive integer<\/p>\n

        Converting Decimal to Standard Form:<\/strong>\u00a0Move decimal RIGHT until A is between 1 and 10 \u2192 n = number of moves \u2192 A \u00d7 10^(-n)<\/p>\n

        Converting Standard Form to Decimal:<\/strong>\u00a0Move decimal LEFT n places<\/p>\n

        Comparing:<\/strong>\u00a0Larger (less negative) exponent = larger number. If exponents same, compare A.<\/p>\n

        Common Small Numbers:<\/strong>
        \n0.1 = 1 \u00d7 10\u207b\u00b9
        \n0.01 = 1 \u00d7 10\u207b\u00b2
        \n0.001 = 1 \u00d7 10\u207b\u00b3
        \n0.0001 = 1 \u00d7 10\u207b\u2074
        \n0.00001 = 1 \u00d7 10\u207b\u2075<\/p>\n

        Real-Life Examples:<\/strong>
        \nRed blood cell: 7 \u00d7 10\u207b\u2076 m
        \nVirus: 1 \u00d7 10\u207b\u2077 m
        \nAtom: 1 \u00d7 10\u207b\u00b9\u2070 m<\/p>\n

        \u00a0<\/p>\n

        <\/p>\n