{"id":29,"date":"2019-06-13T12:00:56","date_gmt":"2019-06-13T16:00:56","guid":{"rendered":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/chapter\/unit-11-exponents-roots-and-scientific-notation\/"},"modified":"2023-11-27T14:19:23","modified_gmt":"2023-11-27T19:19:23","slug":"unit-11-exponents-roots-and-scientific-notation","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/chapter\/unit-11-exponents-roots-and-scientific-notation\/","title":{"raw":"1.5. Exponents and Scientific Notation","rendered":"1.5. Exponents and Scientific Notation"},"content":{"raw":"
\n\n[Latexpage]\n

Exponents<\/h1>\n<\/div>\nExponent review:<\/strong>\u00a0\u00a0an<\/sup><\/em> or BaseExponent<\/sup>\n\n\n\n
Exponential notation<\/strong><\/td>\nExample<\/strong><\/td>\n<\/tr>\n<\/thead>\n<\/table>\n\n\n\n\n\n
Base<\/span> \u00a0 \u00a0 Exponent<\/span><\/td>\n<\/td>\n<\/tr>\n
a<\/em><\/strong><\/span>n<\/strong><\/sup><\/span>= a \u2219 a \u2219 a \u2219 a \u2026 a<\/em><\/td>\n24<\/sup> = 2 \u2219 2 \u2219 2 \u2219 2\u00a0 = 16<\/td>\n<\/tr>\n
Read \u201ca<\/em> to the n<\/em>th\u201d or \u201cthe n<\/em>th power of a<\/em>.\u201d<\/td>\nRead \u201c2 to the 4th.\u201d<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\nProperties of exponents:<\/strong>\n\n\n\n
Name<\/strong><\/td>\nRule<\/strong><\/td>\nExample<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n\n\n\n
Product rule<\/td>\n$a^m\\;a^n=a^{m+n}$<\/td>\n$2^3\\;2^2=2^{3 + 2}=2^5=32$<\/td>\n<\/tr>\n
Quotient rule<\/td>\n$\\frac{a^m}{a^n}=a^{m-n}$<\/td>\n$\\frac{y^4}{y^2}=y^{4-2}=y^2$<\/td>\n<\/tr>\n
Power rule<\/td>\n$(a^m)^n=a^{mn}$\n\n$(a^m \\cdot b^n)^p=a^{mp}\\;b^{np}$\n\n$(\\frac{a^m}{b^n})^p=\\frac{a^{mp}}{b^{np}}$<\/td>\n$(x^3)^2=x^{3\\cdot2}=x^6$\n\n$(t^3 \\cdot s^4)^2=t^{3 \\cdot 2}\\;s^{4 \\cdot 2}=t^6\\;s^8$\n\n$(\\frac{q^2}{p^4})^3=\\frac{q^{2\\cdot3}}{p^{4\\cdot3}}=\\frac{q^6}{p^{12}}$<\/td>\n<\/tr>\n
Negative exponent a-n<\/sup><\/em><\/td>\n$a^{-n}=\\frac{1}{a^n}$<\/td>\n$4^{-2}=\\frac{1}{4^2}=\\frac{1}{16}$<\/td>\n<\/tr>\n
$\\frac{1}{a^{-n}}=a^n$<\/td>\n$\\frac{1}{4^{-2}}=4^2=16$<\/td>\n<\/tr>\n
Zero exponent\u00a0\u00a0 a<\/em>0<\/sup><\/td>\n$a^0=1$<\/td>\n$15^0=1$<\/td>\n<\/tr>\n
One exponent\u00a0\u00a0 <\/em>a<\/em>1<\/sup><\/td>\n$a^1=a$<\/td>\n$7^1=7$\u00a0 \u00a0,\u00a0 \u00a0$1^{13}=1$<\/td>\n<\/tr>\n
Fractional exponent<\/td>\n$a^\\frac{m}{n}=\\sqrt[n]{a^m}$<\/td>\n$15^\\frac{2}{3}=\\sqrt[3]{15^2}$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n
    \n \t
  • Product rule<\/strong>: when multiplying two powers with the same base, keep the base and add the exponents.<\/li>\n<\/ul>\n

    am<\/sup><\/em> a<\/em>n<\/sup> = am<\/sup><\/em> + n<\/em><\/sup>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 an<\/sup><\/em> \u00a0\u00a0or\u00a0 \u00a0BaseExponent<\/sup><\/p>\n\n

    \n\n\n\n\n
    Example<\/strong>:<\/td>\n23 <\/sup><\/strong>22<\/sup><\/strong> = (2 \u00b7 2 \u00b7 2) (2 \u00b7 2) = 25<\/sup> = 32<\/td>\n<\/td>\n<\/tr>\n
    Or<\/td>\n23 <\/sup><\/strong>22 <\/sup><\/strong>= 23 + 2<\/sup> = 25 <\/sup>= 32<\/td>\nA short cut, am<\/sup><\/em> a<\/em>n<\/sup> = am<\/sup><\/em> + n<\/em><\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n \n
      \n \t
    • Quotient rule:<\/strong>\u00a0when dividing two powers with the same base, keep the base and subtract the exponents.<\/li>\n<\/ul>\n

      $\\frac{a^m}{a^n}=a^{m-n}$<\/p>\n\n

      \n\n\n\n\n
      Example<\/strong>:<\/td>\n$\\frac{\\bf2^4}{\\bf2^2}=\\frac{2\\cdot2\\cdot\\bcancel{2\\cdot2}}{\\bcancel{2\\cdot2}}=2^2=4$<\/td>\n<\/td>\n<\/tr>\n
      Or<\/td>\n$\\frac{\\bf2^4}{\\bf2^2}=2^{4-2}=2^2=4$<\/td>\nA short cut, $\\frac{a^m}{a^n}=a^{m-n}$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nThis law can also show that why\u00a0a<\/em>0<\/sup> = 1\u00a0(zero exponent a<\/em>0<\/sup>):\u00a0 $\\frac{a^2}{a^2}=a^{2-2}=a^0=1$\n\n<\/div>\n \n
        \n \t
      • Power rule: <\/strong>when raising an expression to a power, we multiply each exponent inside the parentheses by the power outside the parentheses.<\/li>\n<\/ul>\n

        (am<\/sup><\/em>)<\/span>n<\/sup> = amn<\/sup>,\u00a0 \u00a0 \u00a0 \u00a0 (am<\/sup> \u00b7 bn<\/sup>)p<\/sup> = amp<\/sup> bnp<\/sup>,\u00a0 \u00a0 \u00a0 \u00a0 $(\\frac{a^m}{b^n})^p=\\frac{a^{mp}}{b^{np}}$<\/em><\/p>\n\n

        \n\n\n\n\n
        Example<\/strong>:<\/td>\n(43<\/sup>)2<\/sup><\/strong> = <\/em>(43<\/sup>) (43<\/sup>) = (4 \u00b7 4 \u00b7 4) (4 \u00b7 4 \u00b7 4) = <\/em>46<\/sup> = 4096<\/td>\n<\/td>\n<\/tr>\n
        Or<\/td>\n(43<\/sup>)2<\/sup><\/strong> = <\/em>43 \u2219<\/em> 2<\/sup> = <\/em>46<\/sup> = 4096<\/td>\nA short cut, (am<\/sup><\/em>)n<\/sup> = amn<\/sup><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
        Example<\/strong>:<\/td>\n(2 \u00b7 3<\/strong>)2<\/sup><\/strong> = (2 \u00b7 3) (2 \u00b7 3) = 6 \u2219 6 = 36<\/td>\n<\/td>\n<\/tr>\n
        Or<\/td>\n(2 \u00b7 3<\/strong>)2 <\/sup><\/strong>= 22<\/sup> 32<\/sup> = 4 \u2219 9 = 36<\/td>\nA short cut , (a \u00b7<\/em> b<\/em>)n<\/em><\/sup> = an<\/sup><\/em> bn<\/sup><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
        Example<\/strong>:<\/td>\n$(\\frac{2^2}{3^4})^3=(\\frac{2^2}{3^4})(\\frac{2^2}{3^4})(\\frac{2^2}{3^4})=\\frac{4\\cdot4\\cdot4}{81\\cdot81\\cdot81}=\\frac{64}{531441}$<\/td>\n<\/td>\n<\/tr>\n
        Or<\/td>\n$(\\frac{2^2}{3^4})^3=\\frac{2^{2\\cdot3}}{3^{4\\cdot3}}=\\frac{2^6}{3^{12}}=\\frac{64}{531441}$<\/td>\nA short cut, $(\\frac{a^m}{b^n})^p=\\frac{a^{mp}}{b^{np}}$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n \n
          \n \t
        • Negative exponent:<\/strong> a negative\u00a0exponent\u00a0is the reciprocal of the number with a positive exponent.<\/li>\n<\/ul>\n

          $a^{-n}=\\frac{1}{a^n}$,\u00a0 $\\frac{1}{a^{-n}}=a^n$\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 a<\/em>\u2212n<\/em><\/sup> is the reciprocal of\u00a0an<\/sup><\/em>.<\/p>\n\n

          \n\n\n\n\n
          Example<\/strong>:<\/td>\n$3^{-4}=\\frac{1}{3^4}=\\frac{1}{81}$<\/td>\n$a^{-n}=\\frac{1}{a^n}$<\/td>\n<\/tr>\n
          Example<\/strong>:<\/td>\n$\\frac{1}{3^{-4}}=3^4=81$<\/td>\n$\\frac{1}{a^{-n}}=a^n$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n \n
            \n \t
          • Fractional exponent:<\/strong> a fractional exponent is a different way of writing a radical (i.e. root) sign. The base is first taken to the exponent of\u00a0m<\/em>, then the\u00a0n<\/em>th root is found to obtain the power.<\/li>\n<\/ul>\n

            $a^{\\frac{m}{n}} = {\\sqrt[n]{a}^{m}} = {\\sqrt[n]{a^{m}}}$<\/p>\n\n

            \n\n\n\n
            Example<\/strong>:<\/td>\n$5^{\\frac{3}{2}} = {\\sqrt[2]{5}^{3}} = {\\sqrt[2]{5^{3}}}$<\/td>\n$a^{\\frac{m}{n}} = {\\sqrt[n]{a^{m}}}$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n \n
            \n

            Example 1.5.1<\/p>\n\n<\/header>\n

            \n\nSimplify (do not leave negative exponents in the answer).\n\n\n\n\n\n\n\n\n\n\n\n\n
            1)<\/strong> ${\\bf (-4)^1}=-4$<\/td>\n$a^1=a$<\/td>\n<\/tr>\n
            2) <\/strong>${\\bf (-2345)^0}=1$<\/td>\n$a^0=1$<\/td>\n<\/tr>\n
            3) <\/strong>${\\bf x^2x^3}=x^{2+3}=x^5$<\/td>\n$a^m\\;a^n=a^{m+n}$<\/td>\n<\/tr>\n
            4) <\/strong>${\\bf \\frac{y^6}{y^4}}=y^{6-4}=y^2$<\/td>\n$\\frac{a^m}{a^n}=a^{m-n}$<\/td>\n<\/tr>\n
            5) <\/strong>${\\bf (x^4)^{-3}}=x^{4(-3)}=x^{-12}=\\frac{1}{x^{12}}$<\/td>\n$(a^m)^n=a^{mn}$\u00a0 ,\u00a0 $\\frac{1}{a^{-n}}=a^n$<\/td>\n<\/tr>\n
            6) <\/strong>${\\bf 7b^{-1}}=7\\cdot \\frac{1}{b^1}=\\frac{7}{b}$<\/td>\n$a^{-n}=\\frac{1}{a^n}$\u00a0 ,\u00a0 $a^1=a$<\/td>\n<\/tr>\n
            7) <\/strong>${\\bf (2t^3\\cdot\u00a0 w^2)^4}=2^4 t^{3\\cdot4}\\cdot w^{2\\cdot4}=16t^{12} w^8$<\/td>\n$(a^m \\cdot b^n)^p=a^{mp}\\;b^{np}$<\/td>\n<\/tr>\n
            8) <\/strong>${\\bf \\frac{1}{3^{-2}}}=3^2=9$<\/td>\n$\\frac{1}{a^{-n}}=a^n$<\/td>\n<\/tr>\n
            9) <\/strong>${\\bf \\frac{7x^4y^{-5}}{9^0\\cdot x^2y^3}}=\\frac{7x^{4-2}y^{-5-3}}{1}=7x^2y^{-8}=\\frac{7x^2}{y^8}$<\/td>\n$a^0=1$\u00a0 ,\u00a0 $\\frac{a^m}{a^n}=a^{m-n}$\u00a0 ,\u00a0 $a^{-n}=\\frac{1}{a^n}$<\/td>\n<\/tr>\n
            10) <\/strong>${\\bf (\\frac{e^{-3}f^2}{g^{-2}})^{-2}}=\\frac{e^{(-3)(-2)}f^{2(-2)}}{g^{(-2)(-2)}}=\\frac{e^6f^{-4}}{g^4}=\\frac{e^6}{g^4f^4}$<\/td>\n$(\\frac{a^m}{b^n})^p=\\frac{a^{mp}}{b^{np}}$\u00a0 ,\u00a0 $\\frac{1}{a^{-n}}=a^n$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

            \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/strong><\/p>\n\n

            \u00a0Simplifying Exponential Expressions<\/strong><\/h1>\n
              \n \t
            • Remove parentheses using \u201cpower rule\u201d if necessary. \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em> (a<\/em>m<\/sup> b<\/em>n<\/sup>)p<\/sup> = amp<\/sup><\/em> b<\/em>np<\/sup><\/li>\n \t
            • Regroup coefficients and variables.<\/li>\n \t
            • Use \u201cproduct rule\u201d and \u201cquotient rule\u201d. \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 am<\/sup> an<\/sup> = am + n<\/sup> , <\/em>$\\frac{a^m}{a^n}=a^{m-n}$<\/li>\n \t
            • Simplify.<\/li>\n \t
            • Use the \u201cnegative exponent\u201d rule to make all exponents positive if necessary.<\/li>\n<\/ul>\n \n
              \n

              Example 1.5.2<\/p>\n\n<\/header>\n

              \n\nSimplify.\n\n\n\n\n\n\n\n
              1) <\/strong>$\\bf (3x^3y^2)^2 (2x^{-3}y^{-1})^3 (-248z^{-19})^0$<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
              $=3^2x^{3\\cdot2}y^{2\\cdot2} \\cdot 2^3x^{-3\\cdot3} \\cdot y^{-1\\cdot3}\\cdot1$<\/td>\nRemove brackets.<\/td>\n$(\\frac{a^m}{b^n})^p=\\frac{a^{mp}}{b^{np}}$ , <\/em>$a^0=1$<\/td>\n<\/tr>\n
              $=(3^2\\cdot2^3)(x^6x^{-9})(y^4y^{-3})$<\/td>\nRegroup coefficients and variables.<\/td>\n<\/td>\n<\/tr>\n
              $=72x^{-3}y^1$<\/td>\nSimplify.<\/td>\n$a^m\\;a^n=a^{m+n}$<\/td>\n<\/tr>\n
              $=\\frac{72y}{x^3}$<\/td>\nMake exponent positive.<\/td>\n$a^{-n}=\\frac{1}{a^n}$ , $a^1=a$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n
              2)<\/strong> $\\bf (\\frac{(2x^4)(y^5)}{3x^3y^2})^2$<\/td>\n<\/td>\n$(\\frac{a^m}{b^n})^p=\\frac{a^{mp}}{b^{np}}$<\/td>\n<\/tr>\n
              $=\\frac{(2x^4)^2(y^5)^2}{(3x^3y^2)^2}$<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
              $=\\frac{2^2x^{4\\cdot2}y^{5\\cdot2}}{3^2x^{3\\cdot2}y^{2\\cdot2}}$<\/td>\nRemove brackets.<\/td>\n$(a \\cdot b)^n=a^n\\;b^n$<\/td>\n<\/tr>\n
              $=\\frac{4}{9}\\cdot \\frac{x^8}{x^6}\\cdot \\frac{y^{10}}{y^4}$<\/td>\nRegroup coefficients and variables.<\/td>\n<\/td>\n<\/tr>\n
              $=\\frac{4}{9}x^2y^6$<\/td>\nSimplify.<\/td>\n$\\frac{a^m}{a^n}=a^{m-n}$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n \n
              \n

              Example 1.5.3<\/p>\n\n<\/header>\n

              \n\nEvaluate for\u00a0\u00a0 a<\/em> = 2,\u00a0 \u00a0b<\/em> = 1,\u00a0\u00a0 c<\/em> = -1.\n\n\n\n
              1)<\/strong> ${\\bf (-29a^{-5}b^4c^{-7})^0}=1$<\/td>\n$a^0=1$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
              2)<\/strong> ${\\bf (\\frac{a}{b})^{-4}}=(\\frac{2}{1})^{-4}$<\/td>\nSubstitute 2 for a <\/em>and 1 for b,<\/em><\/td>\n<\/tr>\n
              $=\\frac{2^{-4}}{1^{-4}}=\\frac{1^4}{2^4}=\\frac{1}{16}$<\/td>\n$\\frac{a^m}{a^n}=a^{m-n}$\u00a0 ,\u00a0 $a^{-n}=\\frac{1}{a^n}$\u00a0 ,\u00a0 $\\frac{1}{a^{-n}}=a^n$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n
              3) ${\\bf (a+b-c)^a}=[2+1-(-1)]^2=4^2=16$<\/td>\nSubstitute 2 for a <\/em>and 1 for b, <\/em>and -1 for c.<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
              \n

              <\/h1>\n

              Scientific Notation<\/strong><\/h1>\n<\/div>\nScientific notation <\/strong>is a special format\u00a0to concisely express very large<\/em> and small<\/em> numbers.\n
              \n\nExample<\/strong>:\n\n300,000,000 = 3 \u00d7 108<\/sup> m\/sec. The speed of light.\n\n0.00000000000000000016 = 1.6 \u00d7 10-19<\/sup> C. An electron.\n\n<\/div>\n \n\nScientific notation<\/strong>: a product of a number between 1 and 10<\/span> and a power of 10<\/span>.\n\n\n\n
              Scientific notation<\/strong><\/td>\nExample<\/strong><\/td>\n<\/tr>\n<\/thead>\n<\/table>\n\n\n\n\n
              N<\/em> \u00d7 10\u00b1n<\/em><\/td>\n1 \u2264 N < 10<\/td>\n67504.3 = 6.75043 \u00d7 104<\/sup><\/td>\n<\/tr>\n
              n<\/em> - integer<\/td>\nStandard form<\/td>\nScientific notation<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\u00a0<\/em>\n\nWriting a number in scientific notation:<\/strong>\n\n\n\n
              Step<\/strong><\/td>\nExample<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n
              \n
                \n \t
              • Move the decimal point after<\/em><\/strong> the first nonzero digit<\/em><\/strong>.<\/li>\n<\/ul>\n<\/td>\n
              		\n

              0.007<\/strong>9\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a03<\/strong>7213000<\/p>\n<\/td>\n<\/tr>\n

              \n
                \n \t
              • Determine n<\/em> (the power of 10) by counting the number of places you moved the decimal.<\/li>\n<\/ul>\n<\/td>\n
              		\n

              n<\/em> = 3\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 n<\/em> = 7<\/p>\n<\/td>\n<\/tr>\n

              \n
                \n \t
              • If the decimal point is moved to the right<\/em>:<\/strong> \u00d7 10-n<\/em><\/strong><\/sup><\/li>\n<\/ul>\n<\/td>\n
              		\n

              0.007<\/strong>9 = 7.9 \u00d7 10-3<\/sup><\/p>\n

              3 places to the right.<\/p>\n<\/td>\n<\/tr>\n

              \n
                \n \t
              • If the decimal point is moved to the left<\/em>: <\/strong>\u00d7 10n<\/em><\/strong><\/sup><\/li>\n<\/ul>\n<\/td>\n
              		\n

              3<\/strong>7213000. = 3.7213 \u00d7 107<\/sup><\/p>\n

              7 places to the left.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n

              \n

              Example 1.5.4<\/p>\n\n<\/header>\n

              \n\nWrite in scientific notation.\n\n1)<\/strong>\u00a0\u00a0\u00a0\u00a0 2340000<\/strong> = 2340000. = 2.34\u00d7 106<\/sup>\n\n6 places to the left, \u00d7 10n<\/em><\/sup>\n\n2)\u00a0\u00a0\u00a0\u00a0 0.000000439<\/strong> = 4.39 \u00d7 10-7<\/sup>\n\n7 places to the right, \u00d7 10-n<\/em><\/sup>\n\n<\/div>\n<\/div>\n \n
              \n

              Example 1.5.5<\/p>\n\n<\/header>\n

              \n\nWrite in standard (or ordinary) form.\n\n1)\u00a0 \u00a0 \u00a06.<\/strong>4275 <\/strong>\u00d7<\/strong> 104<\/sup><\/strong> = 64275\n\n2)\u00a0 \u00a0\u00a0<\/strong> 2.9 \u00d7 10-3<\/sup> <\/strong>= 0.0029\n\n<\/div>\n<\/div>\n \n
              \n

              Practice questions<\/h1>\n<\/div>\n1.<\/strong> Evaluate:\n

              a. <\/strong>4x<\/em>2<\/sup> + 5y<\/em>,\u00a0 \u00a0 for x <\/em>= 1, \u00a0\u00a0\u00a0y<\/em> = 4<\/p>\n

              b. <\/strong>(2a<\/em>)3<\/sup> \u2013 3b<\/em>,\u00a0 \u00a0 for a <\/em>= 5, \u00a0\u00a0\u00a0b <\/em>= 6<\/p>\n2.<\/strong> Simplify (do not leave negative exponents in the answer):\n

              a. <\/strong>(-92)1<\/sup><\/p>\n

              b. <\/strong>y<\/em>4<\/sup> y<\/em>3<\/sup><\/p>\n

              c. <\/strong>$\\frac{x^9}{x^6}$<\/p>\n

              d. <\/strong>13a<\/em>-1<\/sup><\/p>\n

              e. <\/strong>(3a<\/em>2<\/sup> \u00b7 b<\/em>3<\/sup>)4<\/sup><\/p>\n

              f.<\/strong> $\\frac{5x^5y^{-6}}{11^0x^3y^4}$<\/p>\n

              g. <\/strong>$(\\frac{u^{-2}v^3}{w^{-4}})^{-3}$\u00a0 \u00a0 \u00a0 \u00a0<\/strong><\/p>\n3.<\/strong> Write in scientific notation:\n

              a.<\/strong> 45,600,000<\/p>\n

              b. <\/strong>0.00000523<\/p>\n4.<\/strong> Write in standard (or ordinary) form:\n

              a.<\/strong> 3.578 \u00d7 103<\/sup><\/p>\n

              b.<\/strong> 4.3 \u00d7 10-5<\/sup><\/p>\n ","rendered":"

              \n

              Exponents<\/h1>\n<\/div>\n

              Exponent review:<\/strong>\u00a0\u00a0an<\/sup><\/em> or BaseExponent<\/sup><\/p>\n\n\n\n
              Exponential notation<\/strong><\/td>\nExample<\/strong><\/td>\n<\/tr>\n<\/thead>\n<\/table>\n\n\n\n\n\n
              Base<\/span> \u00a0 \u00a0 Exponent<\/span><\/td>\n<\/td>\n<\/tr>\n
              a<\/em><\/strong><\/span>n<\/strong><\/sup><\/span>= a \u2219 a \u2219 a \u2219 a \u2026 a<\/em><\/td>\n24<\/sup> = 2 \u2219 2 \u2219 2 \u2219 2\u00a0 = 16<\/td>\n<\/tr>\n
              Read \u201ca<\/em> to the n<\/em>th\u201d or \u201cthe n<\/em>th power of a<\/em>.\u201d<\/td>\nRead \u201c2 to the 4th.\u201d<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

               <\/p>\n

              Properties of exponents:<\/strong><\/p>\n\n\n\n
              Name<\/strong><\/td>\nRule<\/strong><\/td>\nExample<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n\n\n\n
              Product rule<\/td>\n\"a^m\;a^n=a^{m+n}\"<\/td>\n\"2^3\;2^2=2^{3 + 2}=2^5=32\"<\/td>\n<\/tr>\n
              Quotient rule<\/td>\n\"\frac{a^m}{a^n}=a^{m-n}\"<\/td>\n\"\frac{y^4}{y^2}=y^{4-2}=y^2\"<\/td>\n<\/tr>\n
              Power rule<\/td>\n\"(a^m)^n=a^{mn}\"<\/p>\n

              \"(a^m \cdot b^n)^p=a^{mp}\;b^{np}\"<\/p>\n

              \"(\frac{a^m}{b^n})^p=\frac{a^{mp}}{b^{np}}\"<\/td>\n

              		\"(x^3)^2=x^{3\cdot2}=x^6\"<\/p>\n

              \"(t^3 \cdot s^4)^2=t^{3 \cdot 2}\;s^{4 \cdot 2}=t^6\;s^8\"<\/p>\n

              \"(\frac{q^2}{p^4})^3=\frac{q^{2\cdot3}}{p^{4\cdot3}}=\frac{q^6}{p^{12}}\"<\/td>\n<\/tr>\n

              Negative exponent a-n<\/sup><\/em><\/td>\n\"a^{-n}=\frac{1}{a^n}\"<\/td>\n\"4^{-2}=\frac{1}{4^2}=\frac{1}{16}\"<\/td>\n<\/tr>\n
              \"\frac{1}{a^{-n}}=a^n\"<\/td>\n\"\frac{1}{4^{-2}}=4^2=16\"<\/td>\n<\/tr>\n
              Zero exponent\u00a0\u00a0 a<\/em>0<\/sup><\/td>\n\"a^0=1\"<\/td>\n\"15^0=1\"<\/td>\n<\/tr>\n
              One exponent\u00a0\u00a0 <\/em>a<\/em>1<\/sup><\/td>\n\"a^1=a\"<\/td>\n\"7^1=7\"\u00a0 \u00a0,\u00a0 \u00a0\"1^{13}=1\"<\/td>\n<\/tr>\n
              Fractional exponent<\/td>\n\"a^\frac{m}{n}=\sqrt[n]{a^m}\"<\/td>\n\"15^\frac{2}{3}=\sqrt[3]{15^2}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

               <\/p>\n

                \n
              • Product rule<\/strong>: when multiplying two powers with the same base, keep the base and add the exponents.<\/li>\n<\/ul>\n

                am<\/sup><\/em> a<\/em>n<\/sup> = am<\/sup><\/em> + n<\/em><\/sup>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 an<\/sup><\/em> \u00a0\u00a0or\u00a0 \u00a0BaseExponent<\/sup><\/p>\n

                \n\n\n\n\n
                Example<\/strong>:<\/td>\n23 <\/sup><\/strong>22<\/sup><\/strong> = (2 \u00b7 2 \u00b7 2) (2 \u00b7 2) = 25<\/sup> = 32<\/td>\n<\/td>\n<\/tr>\n
                Or<\/td>\n23 <\/sup><\/strong>22 <\/sup><\/strong>= 23 + 2<\/sup> = 25 <\/sup>= 32<\/td>\nA short cut, am<\/sup><\/em> a<\/em>n<\/sup> = am<\/sup><\/em> + n<\/em><\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                 <\/p>\n

                  \n
                • Quotient rule:<\/strong>\u00a0when dividing two powers with the same base, keep the base and subtract the exponents.<\/li>\n<\/ul>\n

                  \"\frac{a^m}{a^n}=a^{m-n}\"<\/p>\n

                  \n\n\n\n\n
                  Example<\/strong>:<\/td>\n\"\frac{\bf2^4}{\bf2^2}=\frac{2\cdot2\cdot\bcancel{2\cdot2}}{\bcancel{2\cdot2}}=2^2=4\"<\/td>\n<\/td>\n<\/tr>\n
                  Or<\/td>\n\"\frac{\bf2^4}{\bf2^2}=2^{4-2}=2^2=4\"<\/td>\nA short cut, \"\frac{a^m}{a^n}=a^{m-n}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                  This law can also show that why\u00a0a<\/em>0<\/sup> = 1\u00a0(zero exponent a<\/em>0<\/sup>):\u00a0 \"\frac{a^2}{a^2}=a^{2-2}=a^0=1\"<\/p>\n<\/div>\n

                   <\/p>\n

                    \n
                  • Power rule: <\/strong>when raising an expression to a power, we multiply each exponent inside the parentheses by the power outside the parentheses.<\/li>\n<\/ul>\n

                    (am<\/sup><\/em>)<\/span>n<\/sup> = amn<\/sup>,\u00a0 \u00a0 \u00a0 \u00a0 (am<\/sup> \u00b7 bn<\/sup>)p<\/sup> = amp<\/sup> bnp<\/sup>,\u00a0 \u00a0 \u00a0 \u00a0 \"(\frac{a^m}{b^n})^p=\frac{a^{mp}}{b^{np}}\"<\/em><\/p>\n

                    \n\n\n\n\n
                    Example<\/strong>:<\/td>\n(43<\/sup>)2<\/sup><\/strong> = <\/em>(43<\/sup>) (43<\/sup>) = (4 \u00b7 4 \u00b7 4) (4 \u00b7 4 \u00b7 4) = <\/em>46<\/sup> = 4096<\/td>\n<\/td>\n<\/tr>\n
                    Or<\/td>\n(43<\/sup>)2<\/sup><\/strong> = <\/em>43 \u2219<\/em> 2<\/sup> = <\/em>46<\/sup> = 4096<\/td>\nA short cut, (am<\/sup><\/em>)n<\/sup> = amn<\/sup><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
                    Example<\/strong>:<\/td>\n(2 \u00b7 3<\/strong>)2<\/sup><\/strong> = (2 \u00b7 3) (2 \u00b7 3) = 6 \u2219 6 = 36<\/td>\n<\/td>\n<\/tr>\n
                    Or<\/td>\n(2 \u00b7 3<\/strong>)2 <\/sup><\/strong>= 22<\/sup> 32<\/sup> = 4 \u2219 9 = 36<\/td>\nA short cut , (a \u00b7<\/em> b<\/em>)n<\/em><\/sup> = an<\/sup><\/em> bn<\/sup><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
                    Example<\/strong>:<\/td>\n\"(\frac{2^2}{3^4})^3=(\frac{2^2}{3^4})(\frac{2^2}{3^4})(\frac{2^2}{3^4})=\frac{4\cdot4\cdot4}{81\cdot81\cdot81}=\frac{64}{531441}\"<\/td>\n<\/td>\n<\/tr>\n
                    Or<\/td>\n\"(\frac{2^2}{3^4})^3=\frac{2^{2\cdot3}}{3^{4\cdot3}}=\frac{2^6}{3^{12}}=\frac{64}{531441}\"<\/td>\nA short cut, \"(\frac{a^m}{b^n})^p=\frac{a^{mp}}{b^{np}}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                     <\/p>\n

                      \n
                    • Negative exponent:<\/strong> a negative\u00a0exponent\u00a0is the reciprocal of the number with a positive exponent.<\/li>\n<\/ul>\n

                      \"a^{-n}=\frac{1}{a^n}\",\u00a0 \"\frac{1}{a^{-n}}=a^n\"\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 a<\/em>\u2212n<\/em><\/sup> is the reciprocal of\u00a0an<\/sup><\/em>.<\/p>\n

                      \n\n\n\n\n
                      Example<\/strong>:<\/td>\n\"3^{-4}=\frac{1}{3^4}=\frac{1}{81}\"<\/td>\n\"a^{-n}=\frac{1}{a^n}\"<\/td>\n<\/tr>\n
                      Example<\/strong>:<\/td>\n\"\frac{1}{3^{-4}}=3^4=81\"<\/td>\n\"\frac{1}{a^{-n}}=a^n\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                       <\/p>\n

                        \n
                      • Fractional exponent:<\/strong> a fractional exponent is a different way of writing a radical (i.e. root) sign. The base is first taken to the exponent of\u00a0m<\/em>, then the\u00a0n<\/em>th root is found to obtain the power.<\/li>\n<\/ul>\n

                        \"a^{\frac{m}{n}} = {\sqrt[n]{a}^{m}} = {\sqrt[n]{a^{m}}}\"<\/p>\n

                        \n\n\n\n
                        Example<\/strong>:<\/td>\n\"5^{\frac{3}{2}} = {\sqrt[2]{5}^{3}} = {\sqrt[2]{5^{3}}}\"<\/td>\n\"a^{\frac{m}{n}} = {\sqrt[n]{a^{m}}}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                         <\/p>\n

                        \n
                        \n

                        Example 1.5.1<\/p>\n<\/header>\n

                        \n

                        Simplify (do not leave negative exponents in the answer).<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n
                        1)<\/strong> \"{\bf (-4)^1}=-4\"<\/td>\n\"a^1=a\"<\/td>\n<\/tr>\n
                        2) <\/strong>\"{\bf (-2345)^0}=1\"<\/td>\n\"a^0=1\"<\/td>\n<\/tr>\n
                        3) <\/strong>\"{\bf x^2x^3}=x^{2+3}=x^5\"<\/td>\n\"a^m\;a^n=a^{m+n}\"<\/td>\n<\/tr>\n
                        4) <\/strong>\"{\bf \frac{y^6}{y^4}}=y^{6-4}=y^2\"<\/td>\n\"\frac{a^m}{a^n}=a^{m-n}\"<\/td>\n<\/tr>\n
                        5) <\/strong>\"{\bf (x^4)^{-3}}=x^{4(-3)}=x^{-12}=\frac{1}{x^{12}}\"<\/td>\n\"(a^m)^n=a^{mn}\"\u00a0 ,\u00a0 \"\frac{1}{a^{-n}}=a^n\"<\/td>\n<\/tr>\n
                        6) <\/strong>\"{\bf 7b^{-1}}=7\cdot \frac{1}{b^1}=\frac{7}{b}\"<\/td>\n\"a^{-n}=\frac{1}{a^n}\"\u00a0 ,\u00a0 \"a^1=a\"<\/td>\n<\/tr>\n
                        7) <\/strong>\"{\bf (2t^3\cdot w^2)^4}=2^4 t^{3\cdot4}\cdot w^{2\cdot4}=16t^{12} w^8\"<\/td>\n\"(a^m \cdot b^n)^p=a^{mp}\;b^{np}\"<\/td>\n<\/tr>\n
                        8) <\/strong>\"{\bf \frac{1}{3^{-2}}}=3^2=9\"<\/td>\n\"\frac{1}{a^{-n}}=a^n\"<\/td>\n<\/tr>\n
                        9) <\/strong>\"{\bf \frac{7x^4y^{-5}}{9^0\cdot x^2y^3}}=\frac{7x^{4-2}y^{-5-3}}{1}=7x^2y^{-8}=\frac{7x^2}{y^8}\"<\/td>\n\"a^0=1\"\u00a0 ,\u00a0 \"\frac{a^m}{a^n}=a^{m-n}\"\u00a0 ,\u00a0 \"a^{-n}=\frac{1}{a^n}\"<\/td>\n<\/tr>\n
                        10) <\/strong>\"{\bf (\frac{e^{-3}f^2}{g^{-2}})^{-2}}=\frac{e^{(-3)(-2)}f^{2(-2)}}{g^{(-2)(-2)}}=\frac{e^6f^{-4}}{g^4}=\frac{e^6}{g^4f^4}\"<\/td>\n\"(\frac{a^m}{b^n})^p=\frac{a^{mp}}{b^{np}}\"\u00a0 ,\u00a0 \"\frac{1}{a^{-n}}=a^n\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

                        \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/strong><\/p>\n

                        \u00a0Simplifying Exponential Expressions<\/strong><\/h1>\n
                          \n
                        • Remove parentheses using \u201cpower rule\u201d if necessary. \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em> (a<\/em>m<\/sup> b<\/em>n<\/sup>)p<\/sup> = amp<\/sup><\/em> b<\/em>np<\/sup><\/li>\n
                        • Regroup coefficients and variables.<\/li>\n
                        • Use \u201cproduct rule\u201d and \u201cquotient rule\u201d. \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 am<\/sup> an<\/sup> = am + n<\/sup> , <\/em>\"\frac{a^m}{a^n}=a^{m-n}\"<\/li>\n
                        • Simplify.<\/li>\n
                        • Use the \u201cnegative exponent\u201d rule to make all exponents positive if necessary.<\/li>\n<\/ul>\n

                           <\/p>\n

                          \n
                          \n

                          Example 1.5.2<\/p>\n<\/header>\n

                          \n

                          Simplify.<\/p>\n\n\n\n\n\n\n\n
                          1) <\/strong>\"\bf (3x^3y^2)^2 (2x^{-3}y^{-1})^3 (-248z^{-19})^0\"<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                          \"=3^2x^{3\cdot2}y^{2\cdot2} \cdot 2^3x^{-3\cdot3} \cdot y^{-1\cdot3}\cdot1\"<\/td>\nRemove brackets.<\/td>\n\"(\frac{a^m}{b^n})^p=\frac{a^{mp}}{b^{np}}\" , <\/em>\"a^0=1\"<\/td>\n<\/tr>\n
                          \"=(3^2\cdot2^3)(x^6x^{-9})(y^4y^{-3})\"<\/td>\nRegroup coefficients and variables.<\/td>\n<\/td>\n<\/tr>\n
                          \"=72x^{-3}y^1\"<\/td>\nSimplify.<\/td>\n\"a^m\;a^n=a^{m+n}\"<\/td>\n<\/tr>\n
                          \"=\frac{72y}{x^3}\"<\/td>\nMake exponent positive.<\/td>\n\"a^{-n}=\frac{1}{a^n}\" , \"a^1=a\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n
                          2)<\/strong> \"\bf (\frac{(2x^4)(y^5)}{3x^3y^2})^2\"<\/td>\n<\/td>\n\"(\frac{a^m}{b^n})^p=\frac{a^{mp}}{b^{np}}\"<\/td>\n<\/tr>\n
                          \"=\frac{(2x^4)^2(y^5)^2}{(3x^3y^2)^2}\"<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                          \"=\frac{2^2x^{4\cdot2}y^{5\cdot2}}{3^2x^{3\cdot2}y^{2\cdot2}}\"<\/td>\nRemove brackets.<\/td>\n\"(a \cdot b)^n=a^n\;b^n\"<\/td>\n<\/tr>\n
                          \"=\frac{4}{9}\cdot \frac{x^8}{x^6}\cdot \frac{y^{10}}{y^4}\"<\/td>\nRegroup coefficients and variables.<\/td>\n<\/td>\n<\/tr>\n
                          \"=\frac{4}{9}x^2y^6\"<\/td>\nSimplify.<\/td>\n\"\frac{a^m}{a^n}=a^{m-n}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

                           <\/p>\n

                          \n
                          \n

                          Example 1.5.3<\/p>\n<\/header>\n

                          \n

                          Evaluate for\u00a0\u00a0 a<\/em> = 2,\u00a0 \u00a0b<\/em> = 1,\u00a0\u00a0 c<\/em> = -1.<\/p>\n\n\n\n
                          1)<\/strong> \"{\bf (-29a^{-5}b^4c^{-7})^0}=1\"<\/td>\n\"a^0=1\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
                          2)<\/strong> \"{\bf (\frac{a}{b})^{-4}}=(\frac{2}{1})^{-4}\"<\/td>\nSubstitute 2 for a <\/em>and 1 for b,<\/em><\/td>\n<\/tr>\n
                          \"=\frac{2^{-4}}{1^{-4}}=\frac{1^4}{2^4}=\frac{1}{16}\"<\/td>\n\"\frac{a^m}{a^n}=a^{m-n}\"\u00a0 ,\u00a0 \"a^{-n}=\frac{1}{a^n}\"\u00a0 ,\u00a0 \"\frac{1}{a^{-n}}=a^n\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n
                          3) \"{\bf (a+b-c)^a}=[2+1-(-1)]^2=4^2=16\"<\/td>\nSubstitute 2 for a <\/em>and 1 for b, <\/em>and -1 for c.<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
                          \n

                          <\/h1>\n

                          Scientific Notation<\/strong><\/h1>\n<\/div>\n

                          Scientific notation <\/strong>is a special format\u00a0to concisely express very large<\/em> and small<\/em> numbers.<\/p>\n

                          \n

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

                          300,000,000 = 3 \u00d7 108<\/sup> m\/sec. The speed of light.<\/p>\n

                          0.00000000000000000016 = 1.6 \u00d7 10-19<\/sup> C. An electron.<\/p>\n<\/div>\n

                           <\/p>\n

                          Scientific notation<\/strong>: a product of a number between 1 and 10<\/span> and a power of 10<\/span>.<\/p>\n\n\n\n
                          Scientific notation<\/strong><\/td>\nExample<\/strong><\/td>\n<\/tr>\n<\/thead>\n<\/table>\n\n\n\n\n
                          N<\/em> \u00d7 10\u00b1n<\/em><\/td>\n1 \u2264 N < 10<\/td>\n67504.3 = 6.75043 \u00d7 104<\/sup><\/td>\n<\/tr>\n
                          n<\/em> – integer<\/td>\nStandard form<\/td>\nScientific notation<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                          \u00a0<\/em><\/p>\n

                          Writing a number in scientific notation:<\/strong><\/p>\n\n\n\n
                          Step<\/strong><\/td>\nExample<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n
                          \n
                            \n
                          • Move the decimal point after<\/em><\/strong> the first nonzero digit<\/em><\/strong>.<\/li>\n<\/ul>\n<\/td>\n
                          		\n

                          0.007<\/strong>9\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a03<\/strong>7213000<\/p>\n<\/td>\n<\/tr>\n

                          \n
                            \n
                          • Determine n<\/em> (the power of 10) by counting the number of places you moved the decimal.<\/li>\n<\/ul>\n<\/td>\n
                          		\n

                          n<\/em> = 3\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 n<\/em> = 7<\/p>\n<\/td>\n<\/tr>\n

                          \n
                            \n
                          • If the decimal point is moved to the right<\/em>:<\/strong> \u00d7 10–n<\/em><\/strong><\/sup><\/li>\n<\/ul>\n<\/td>\n
                          		\n

                          0.007<\/strong>9 = 7.9 \u00d7 10-3<\/sup><\/p>\n

                          3 places to the right.<\/p>\n<\/td>\n<\/tr>\n

                          \n
                            \n
                          • If the decimal point is moved to the left<\/em>: <\/strong>\u00d7 10n<\/em><\/strong><\/sup><\/li>\n<\/ul>\n<\/td>\n
                          		\n

                          3<\/strong>7213000. = 3.7213 \u00d7 107<\/sup><\/p>\n

                          7 places to the left.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                           <\/p>\n

                          \n
                          \n

                          Example 1.5.4<\/p>\n<\/header>\n

                          \n

                          Write in scientific notation.<\/p>\n

                          1)<\/strong>\u00a0\u00a0\u00a0\u00a0 2340000<\/strong> = 2340000. = 2.34\u00d7 106<\/sup><\/p>\n

                          6 places to the left, \u00d7 10n<\/em><\/sup><\/p>\n

                          2)\u00a0\u00a0\u00a0\u00a0 0.000000439<\/strong> = 4.39 \u00d7 10-7<\/sup><\/p>\n

                          7 places to the right, \u00d7 10-n<\/em><\/sup><\/p>\n<\/div>\n<\/div>\n

                           <\/p>\n

                          \n
                          \n

                          Example 1.5.5<\/p>\n<\/header>\n

                          \n

                          Write in standard (or ordinary) form.<\/p>\n

                          1)\u00a0 \u00a0 \u00a06.<\/strong>4275 <\/strong>\u00d7<\/strong> 104<\/sup><\/strong> = 64275<\/p>\n

                          2)\u00a0 \u00a0\u00a0<\/strong> 2.9 \u00d7 10-3<\/sup> <\/strong>= 0.0029<\/p>\n<\/div>\n<\/div>\n

                           <\/p>\n

                          \n

                          Practice questions<\/h1>\n<\/div>\n

                          1.<\/strong> Evaluate:<\/p>\n

                          a. <\/strong>4x<\/em>2<\/sup> + 5y<\/em>,\u00a0 \u00a0 for x <\/em>= 1, \u00a0\u00a0\u00a0y<\/em> = 4<\/p>\n

                          b. <\/strong>(2a<\/em>)3<\/sup> \u2013 3b<\/em>,\u00a0 \u00a0 for a <\/em>= 5, \u00a0\u00a0\u00a0b <\/em>= 6<\/p>\n

                          2.<\/strong> Simplify (do not leave negative exponents in the answer):<\/p>\n

                          a. <\/strong>(-92)1<\/sup><\/p>\n

                          b. <\/strong>y<\/em>4<\/sup> y<\/em>3<\/sup><\/p>\n

                          c. <\/strong>\"\frac{x^9}{x^6}\"<\/p>\n

                          d. <\/strong>13a<\/em>-1<\/sup><\/p>\n

                          e. <\/strong>(3a<\/em>2<\/sup> \u00b7 b<\/em>3<\/sup>)4<\/sup><\/p>\n

                          f.<\/strong> \"\frac{5x^5y^{-6}}{11^0x^3y^4}\"<\/p>\n

                          g. <\/strong>\"(\frac{u^{-2}v^3}{w^{-4}})^{-3}\"\u00a0 \u00a0 \u00a0 \u00a0<\/strong><\/p>\n

                          3.<\/strong> Write in scientific notation:<\/p>\n

                          a.<\/strong> 45,600,000<\/p>\n

                          b. <\/strong>0.00000523<\/p>\n

                          4.<\/strong> Write in standard (or ordinary) form:<\/p>\n

                          a.<\/strong> 3.578 \u00d7 103<\/sup><\/p>\n

                          b.<\/strong> 4.3 \u00d7 10-5<\/sup><\/p>\n

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