document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209558>Question 1209558</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a) If 7^sin²x - 7^cos²x = 8<BR>\n" );
document.write( "find the value of x.\r<BR>\n" );
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document.write( "b) Let m be a given real number, Find all complex numbers x such that,<BR>\n" );
document.write( "([x/(x+1)]² + [x/(x-1)]² = m² + m\r<BR>\n" );
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document.write( "c) ab = ½, bc = ⅓, ac = 1/6,<BR>\n" );
document.write( "find (1/a²) + (1/b²) + (1/c²)\r<BR>\n" );
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document.write( "d) 2^x - 3^y = 5<BR>\n" );
document.write( "2^(x+2) + 3^(y+2) = 59. <BR>\n" );
document.write( "find xy\r<BR>\n" );
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document.write( "e) find m, <BR>\n" );
document.write( "9^4^m = 4^9^m\r<BR>\n" );
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document.write( "f) (7^log₈x)×(x^log₉x) = 3969, <BR>\n" );
document.write( "find x.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849496 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849496\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve these problems:\r<BR>\n" );
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document.write( "**a) 7^(sin²x) - 7^(cos²x) = 8**\r<BR>\n" );
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document.write( "1.  Use the identity cos²x = 1 - sin²x:\r<BR>\n" );
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document.write( "7^(sin²x) - 7^(1-sin²x) = 8\r<BR>\n" );
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document.write( "2.  Rewrite the second term:\r<BR>\n" );
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document.write( "7^(sin²x) - 7 / 7^(sin²x) = 8\r<BR>\n" );
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document.write( "3.  Let y = 7^(sin²x):\r<BR>\n" );
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document.write( "y - 7/y = 8\r<BR>\n" );
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document.write( "4.  Multiply by y:\r<BR>\n" );
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document.write( "y² - 7 = 8y\r<BR>\n" );
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document.write( "5.  Rearrange into a quadratic equation:\r<BR>\n" );
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document.write( "y² - 8y - 7 = 0\r<BR>\n" );
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document.write( "6.  Factor:\r<BR>\n" );
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document.write( "(y - 7)(y + 1) = 0\r<BR>\n" );
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document.write( "7.  Solve for y:\r<BR>\n" );
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document.write( "y = 7 or y = -1\r<BR>\n" );
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document.write( "Since 7^(sin²x) must be positive, y = -1 is not a valid solution.  So, y = 7.\r<BR>\n" );
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document.write( "8.  Substitute back:\r<BR>\n" );
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document.write( "7^(sin²x) = 7¹\r<BR>\n" );
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document.write( "sin²x = 1\r<BR>\n" );
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document.write( "9.  Solve for x:\r<BR>\n" );
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document.write( "sin x = ±1<BR>\n" );
document.write( "x = (2n + 1)π/2, where n is an integer.\r<BR>\n" );
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document.write( "**b) [x/(x+1)]² + [x/(x-1)]² = m² + m**\r<BR>\n" );
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document.write( "1.  Simplify the left side:\r<BR>\n" );
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document.write( "x²/(x²+2x+1) + x²/(x²-2x+1) = m² + m\r<BR>\n" );
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document.write( "2. Find a common denominator and combine the fractions:\r<BR>\n" );
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document.write( "[x²(x²-2x+1) + x²(x²+2x+1)] / [(x²+2x+1)(x²-2x+1)] = m² + m\r<BR>\n" );
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document.write( "[x⁴-2x³+x² + x⁴+2x³+x²] / (x⁴-2x²+1) = m² + m\r<BR>\n" );
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document.write( "(2x⁴ + 2x²) / (x⁴ - 2x² + 1) = m² + m\r<BR>\n" );
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document.write( "2x²(x² + 1) / (x² - 1)² = m(m + 1)\r<BR>\n" );
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document.write( "This equation is a bit complex.  Without a specific value for 'm', it's difficult to simplify further to directly solve for x.  The best approach would be to substitute a given value for m and then try to solve for x.\r<BR>\n" );
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document.write( "**c) ab = ½, bc = ⅓, ac = 1/6**\r<BR>\n" );
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document.write( "1. Multiply the three equations together:\r<BR>\n" );
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document.write( "(ab)(bc)(ac) = (1/2)(1/3)(1/6)<BR>\n" );
document.write( "a²b²c² = 1/36\r<BR>\n" );
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document.write( "2. Take the square root of both sides:\r<BR>\n" );
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document.write( "abc = ±1/6\r<BR>\n" );
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document.write( "3. Divide (abc = ±1/6) by each of the original equations to get the reciprocals:\r<BR>\n" );
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document.write( "1/c = (abc)/(ab) = (±1/6)/(1/2) = ±1/3, so c² = 9<BR>\n" );
document.write( "1/a = (abc)/(bc) = (±1/6)/(1/3) = ±1/2, so a² = 4<BR>\n" );
document.write( "1/b = (abc)/(ac) = (±1/6)/(1/6) = ±1, so b² = 1\r<BR>\n" );
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document.write( "4. Calculate the sum of the reciprocals squared:\r<BR>\n" );
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document.write( "(1/a²) + (1/b²) + (1/c²) = 1/4 + 1 + 9 = 10.25 = 41/4\r<BR>\n" );
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document.write( "**d) 2^x - 3^y = 5 and 2^(x+2) + 3^(y+2) = 59**\r<BR>\n" );
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document.write( "1. Rewrite the second equation:\r<BR>\n" );
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document.write( "4 * 2^x + 9 * 3^y = 59\r<BR>\n" );
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document.write( "2. Let u = 2^x and v = 3^y. The system becomes:\r<BR>\n" );
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document.write( "u - v = 5<BR>\n" );
document.write( "4u + 9v = 59\r<BR>\n" );
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document.write( "3. Solve for u and v: From the first equation, u = v + 5. Substituting this into the second equation:\r<BR>\n" );
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document.write( "4(v + 5) + 9v = 59<BR>\n" );
document.write( "13v + 20 = 59<BR>\n" );
document.write( "13v = 39<BR>\n" );
document.write( "v = 3\r<BR>\n" );
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document.write( "Then, u = v + 5 = 3 + 5 = 8.\r<BR>\n" );
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document.write( "4. Substitute back to find x and y:\r<BR>\n" );
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document.write( "2^x = 8 = 2³  => x = 3<BR>\n" );
document.write( "3^y = 3 = 3¹  => y = 1\r<BR>\n" );
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document.write( "5. Find xy:\r<BR>\n" );
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document.write( "xy = 3 * 1 = 3\r<BR>\n" );
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document.write( "**e) 9^(4^m) = 4^(9^m)**\r<BR>\n" );
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document.write( "1. Take the logarithm of both sides (any base, but let's use the natural log):\r<BR>\n" );
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document.write( "ln(9^(4^m)) = ln(4^(9^m))\r<BR>\n" );
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document.write( "2. Use the logarithm power rule:\r<BR>\n" );
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document.write( "4^m * ln(9) = 9^m * ln(4)\r<BR>\n" );
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document.write( "3. Rewrite ln(9) as 2ln(3) and ln(4) as 2ln(2):\r<BR>\n" );
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document.write( "4^m * 2ln(3) = 9^m * 2ln(2)\r<BR>\n" );
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document.write( "4. Simplify:\r<BR>\n" );
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document.write( "4^m * ln(3) = 9^m * ln(2)\r<BR>\n" );
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document.write( "5. Rearrange:\r<BR>\n" );
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document.write( "4^m / 9^m = ln(2) / ln(3)\r<BR>\n" );
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document.write( "(2²/3²)^m = ln(2) / ln(3)\r<BR>\n" );
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document.write( "(2/3)^(2m) = log₃2\r<BR>\n" );
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document.write( "2m * ln(2/3) = ln(log₃2)\r<BR>\n" );
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document.write( "m = ln(log₃2) / (2 * ln(2/3))\r<BR>\n" );
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document.write( "**f) (7^(log₈x)) * (x^(log₉x)) = 3969**\r<BR>\n" );
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document.write( "1. Take the logarithm base 8 of both sides:\r<BR>\n" );
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document.write( "log₈[(7^(log₈x)) * (x^(log₉x))] = log₈3969\r<BR>\n" );
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document.write( "2. Use logarithm properties:\r<BR>\n" );
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document.write( "log₈(7^(log₈x)) + log₈(x^(log₉x)) = log₈3969\r<BR>\n" );
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document.write( "(log₈x)(log₈7) + (log₉x)(log₈x) = log₈3969\r<BR>\n" );
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document.write( "3. Let y = log₈x:\r<BR>\n" );
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document.write( "y * log₈7 + (log₉x) * y = log₈3969\r<BR>\n" );
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document.write( "y(log₈7 + log₉x) = log₈3969\r<BR>\n" );
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document.write( "We also know that 3969 = 63² = 9*7*9*7 = 3⁴ * 7²\r<BR>\n" );
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document.write( "log₈3969 = log₈(3⁴ * 7²) = 4log₈3 + 2log₈7\r<BR>\n" );
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document.write( "y(log₈7 + log₉x) = 4log₈3 + 2log₈7\r<BR>\n" );
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document.write( "This equation is still quite complex. It's likely there's a clever substitution or manipulation I'm missing to simplify it further and solve directly for x. Numerical methods or approximations might be necessary.<BR>\n" );
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