document.write( "Question 65419: Rational and Radical Expressions \r
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\n" ); document.write( "\n" ); document.write( "1. Simplify the rational expression:
\n" ); document.write( "x2 – 6x – 7
\n" ); document.write( "———————————
\n" ); document.write( " x2 – 1\r
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\n" ); document.write( "\n" ); document.write( "2. Divide:
\n" ); document.write( "x2 – 3x + 2 ÷ x2 – 4
\n" ); document.write( " ——————————— —————————
\n" ); document.write( " 8x – 8 5x + 10\r
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\n" ); document.write( "\n" ); document.write( "3. Simplify:
\n" ); document.write( "√[100x2y16z8]\r
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\n" ); document.write( "\n" ); document.write( "4. Perform the indicated operations:
\n" ); document.write( "2√[72] + √[32] – 4√[18]\r
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\n" ); document.write( "\n" ); document.write( "5. Multiply:
\n" ); document.write( "(√[3] + 4√[5])(2√[3] – √[5])\r
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\n" ); document.write( "\n" ); document.write( "6. Rationalize the denominator:
\n" ); document.write( " 3
\n" ); document.write( "———————————
\n" ); document.write( "√[6] – √[3]\r
\n" ); document.write( "\n" ); document.write( "thank you so much\r
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Algebra.Com's Answer #45963 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
Rational and Radical Expressions \r\n" );
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document.write( "1.  Simplify the rational expression:\r\n" );
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document.write( "x² – 6x – 7\r\n" );
document.write( "———————————                   \r\n" );
document.write( "  x² – 1\r\n" );
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document.write( "Factor top and bottom:\r\n" );
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document.write( " (x - 7)(x + 1)\r\n" );
document.write( "————————————————                   \r\n" );
document.write( " (x - 1)(x + 1)\r\n" );
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document.write( "Cancel the (x + 1)'s\r\n" );
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document.write( "           1\r\n" );
document.write( " (x - 7)(x + 1)\r\n" );
document.write( "————————————————                   \r\n" );
document.write( " (x - 1)(x + 1)\r\n" );
document.write( "           1\r\n" );
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document.write( "and all that's left is\r\n" );
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document.write( " x - 7\r\n" );
document.write( "———————                   \r\n" );
document.write( " x - 1 \r\n" );
document.write( "            \r\n" );
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document.write( "2.  Divide:\r\n" );
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document.write( " x² – 3x + 2      x² – 4\r\n" );
document.write( "————————————— ÷ —————————— \r\n" );
document.write( "   8x – 8        5x + 10\r\n" );
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document.write( "Invert the second fraction and change ÷ to ·\r\n" );
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document.write( " x² – 3x + 2      5x + 10\r\n" );
document.write( "————————————— · —————————— \r\n" );
document.write( "   8x – 8          x² - 4\r\n" );
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document.write( "Factor all numerators and denominators:\r\n" );
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document.write( " (x - 2)(x - 1)        5(x + 2)\r\n" );
document.write( "———————————————— · ———————————————— \r\n" );
document.write( "   8(x – 1)         (x - 2)(x + 2)\r\n" );
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document.write( "Indicate the multiplication of all factors\r\n" );
document.write( "in the numerator and denominator all as\r\n" );
document.write( "one fraction:\r\n" );
document.write( "\r\n" );
document.write( "  5(x - 2)(x - 1)(x + 2)\r\n" );
document.write( "—————————————————————————— \r\n" );
document.write( "  8(x – 1)(x - 2)(x + 2)\r\n" );
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document.write( "Cancel the (x - 2)'s\r\n" );
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document.write( "      1        \r\n" );
document.write( "  5(x - 2)(x - 1)(x + 2)\r\n" );
document.write( "—————————————————————————— \r\n" );
document.write( "  8(x – 1)(x - 2)(x + 2)\r\n" );
document.write( "             1\r\n" );
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document.write( "      1      1  \r\n" );
document.write( "  5(x - 2)(x - 1)(x + 2)\r\n" );
document.write( "—————————————————————————— \r\n" );
document.write( "  8(x – 1)(x - 2)(x + 2)\r\n" );
document.write( "      1      1\r\n" );
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document.write( "Cancel the (x + 2)'s\r\n" );
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document.write( "      1      1      1 \r\n" );
document.write( "  5(x - 2)(x - 1)(x + 2)\r\n" );
document.write( "—————————————————————————— \r\n" );
document.write( "  8(x – 1)(x - 2)(x + 2)\r\n" );
document.write( "      1      1      1\r\n" );
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document.write( "all that's left is\r\n" );
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document.write( "       5\r\n" );
document.write( "      ———\r\n" );
document.write( "       8\r\n" );
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document.write( "3.  Simplify:\r\n" );
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document.write( " __________\r\n" );
document.write( "√100x2y16z8 \r\n" );
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document.write( "Write the 10 as 102\r\n" );
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document.write( "To take the square root of an even power,\r\n" );
document.write( "divide the exponent by 2:\r\n" );
document.write( " __________\r\n" );
document.write( "√102x22y16z8\r\n" );
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document.write( "102÷2x2÷2y16÷2z8÷2\r\n" );
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document.write( "101x1y8z4\r\n" );
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document.write( "Erase the two 1 exponents\r\n" );
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document.write( "10xy8z4\r\n" );
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document.write( "4.  Perform the indicated operations:\r\n" );
document.write( " __    __    __\r\n" );
document.write( "Ö72 + Ö32 – Ö18\r\n" );
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document.write( "72 = 8·9 = 4·2·3·3 = 2·2·2·3·3\r\n" );
document.write( "32 = 4·8 = 2·2·4·2 = 2·2·2·2·2\r\n" );
document.write( "18 = 2·9 = 2·3·3\r\n" );
document.write( " _________    _________    _____\r\n" );
document.write( "Ö2·2·2·3·3 + Ö2·2·2·2·2 - Ö2·3·3\r\n" );
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document.write( "Group like factor into pairs:\r\n" );
document.write( " _____________    _____________    _______\r\n" );
document.write( "Ö(2·2)·2·(3·3) + Ö(2·2)·(2·2)·2 - Ö2·(3·3)   \r\n" );
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document.write( "Each pair of like factors comes out in front\r\n" );
document.write( "of the square root radical as a single factor.\r\n" );
document.write( "That is, in the first radical the (2·2) and\r\n" );
document.write( "the (3·3) come out in front of the radical as\r\n" );
document.write( "as single factors multiplied 2·3, and the\r\n" );
document.write( "unpaired 2 stays under the radical.\r\n" );
document.write( "            _       _     _\r\n" );
document.write( "        2·3Ö2 + 2·2Ö2 - 3Ö2\r\n" );
document.write( "            _     _     _\r\n" );
document.write( "          6Ö2 + 4Ö2 - 3Ö2\r\n" );
document.write( "            _\r\n" );
document.write( "Factor out Ö2\r\n" );
document.write( "            _\r\n" );
document.write( "           Ö2(6 + 4 - 3)\r\n" );
document.write( "            _ \r\n" );
document.write( "           Ö2(7)\r\n" );
document.write( "              _\r\n" );
document.write( "            7Ö2\r\n" );
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document.write( "5.  Multiply:\r\n" );
document.write( "  _    _    _    _\r\n" );
document.write( "(Ö3 + Ö5)(2Ö3 – Ö5)\r\n" );
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document.write( "Use FOIL\r\n" );
document.write( "  _    _      _   _      _    _      _   _\r\n" );
document.write( "(Ö3)(2Ö3) – (Ö3)(Ö5) + (Ö5)(2Ö3) – (Ö5)(Ö5)\r\n" );
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document.write( "Multply under the radicals:\r\n" );
document.write( "  ___    ___    ___    ___\r\n" );
document.write( "2Ö3·3 - Ö3·5 + 2Ö5·3 - Ö5·5\r\n" );
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document.write( "Pair like factors under the 1st and 4th radicals\r\n" );
document.write( "Multiply factors under the middle two radicals\r\n" );
document.write( "which have no pairs of like factors:\r\n" );
document.write( "  _____    __    __    _____   \r\n" );
document.write( "2Ö(3·3) - Ö15 + 2Ö15 - Ö(5·5) \r\n" );
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document.write( "The pair of 3's comes out eliminating the radical\r\n" );
document.write( "in the first term. The middle two terms combine,\r\n" );
document.write( "and the pair of 5's comes out eliminating the\r\n" );
document.write( "radical in the 4th term:\r\n" );
document.write( "       __\r\n" );
document.write( "2·3 + Ö15 - 5\r\n" );
document.write( "      __\r\n" );
document.write( " 6 + Ö15 - 5\r\n" );
document.write( "        __\r\n" );
document.write( "   1 + Ö15\r\n" );
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document.write( "6.  Rationalize the denominator:\r\n" );
document.write( "         3\r\n" );
document.write( "    ———————————  \r\n" );
document.write( "      Ö6 – Ö3\r\n" );
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document.write( "Form the two term conjugate of the denominator.\r\n" );
document.write( "1. Its first term is_the same as the first term of  \r\n" );
document.write( "   the denominator Ö6\r\n" );
document.write( "2. Its second term is the second term of the   _\r\n" );
document.write( "   denominator with its sign changed, or_Ö6 + Ö3\r\n" );
document.write( "3. So the conjugate of Ö6 - Ö3 is Ö6 + Ö3\r\n" );
document.write( "4. Put the conjugate over itself\r\n" );
document.write( "       _    _\r\n" );
document.write( "      Ö6 + Ö3\r\n" );
document.write( "    ———————————  \r\n" );
document.write( "      Ö6 + Ö3\r\n" );
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document.write( "which just equals 1, so we can then multiply the\r\n" );
document.write( "fraction to be rationalized by this fraction without\r\n" );
document.write( "changing its value, since multiplication by 1 does\r\n" );
document.write( "no change the value of an expression\r\n" );
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document.write( "                     _    _\r\n" );
document.write( "         3          Ö6 + Ö3\r\n" );
document.write( "    ——————————— · ———————————\r\n" );
document.write( "      Ö6 – Ö3       Ö6 + Ö3\r\n" );
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document.write( "Indicate the multiplication of numerators and \r\n" );
document.write( "denominators, all as one fraction:\r\n" );
document.write( "            _    _\r\n" );
document.write( "         3(Ö6 + Ö3)\r\n" );
document.write( "     ————————————————————\r\n" );
document.write( "      (Ö6 – Ö3)(Ö6 + Ö3)\r\n" );
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document.write( "FOIL out the bottom:\r\n" );
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document.write( "               _    _\r\n" );
document.write( "            3(Ö6 + Ö3)\r\n" );
document.write( "     ———————————————————————————\r\n" );
document.write( "      Ö6Ö6 + Ö3Ö6 - Ö6Ö3 - Ö3Ö3\r\n" );
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document.write( "               _    _\r\n" );
document.write( "            3(Ö6 + Ö3)\r\n" );
document.write( "     ———————————————————————\r\n" );
document.write( "      Ö36 + Ö18 - Ö18 - Ö9\r\n" );
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document.write( "               _    _\r\n" );
document.write( "            3(Ö6 + Ö3)\r\n" );
document.write( "     ———————————————————————\r\n" );
document.write( "        6 + Ö18 - Ö18 - 3\r\n" );
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document.write( "The middle terms cancel\r\n" );
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document.write( "               _    _\r\n" );
document.write( "            3(Ö6 + Ö3)\r\n" );
document.write( "           ————————————\r\n" );
document.write( "               6 - 3\r\n" );
document.write( "\r\n" );
document.write( "             \r\n" );
document.write( "               _    _\r\n" );
document.write( "            3(Ö6 + Ö3)\r\n" );
document.write( "           ————————————\r\n" );
document.write( "                 3\r\n" );
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document.write( "Cancel the 3's\r\n" );
document.write( "                  \r\n" );
document.write( "            1  _    _\r\n" );
document.write( "            3(Ö6 + Ö3)\r\n" );
document.write( "           ————————————\r\n" );
document.write( "                 3\r\n" );
document.write( "                 1 \r\n" );
document.write( "               _    _\r\n" );
document.write( "              Ö6 + Ö3\r\n" );
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document.write( "Edwin
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