document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209806>Question 1209806</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose x⁵y¹⁷=r and x²y⁷=s, <BR>\n" );
document.write( "If x=r^a/s^b and y=s^c/r^d, <BR>\n" );
document.write( "find a + b + c + d </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850625 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=850625\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Express x and y in terms of r and s**\r<BR>\n" );
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document.write( "Given:<BR>\n" );
document.write( "* x⁵y¹⁷ = r  ...(1)<BR>\n" );
document.write( "* x²y⁷ = s   ...(2)\r<BR>\n" );
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document.write( "We want to find x and y in the form:<BR>\n" );
document.write( "* x = r^a / s^b<BR>\n" );
document.write( "* y = s^c / r^d\r<BR>\n" );
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document.write( "**2. Solve for x and y**\r<BR>\n" );
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document.write( "We need to eliminate one of the variables to express the other in terms of r and s.\r<BR>\n" );
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document.write( "* **Eliminate y:**<BR>\n" );
document.write( "    * Raise equation (2) to the power of 17: (x²y⁷)¹⁷ = s¹⁷  => x³⁴y¹¹⁹ = s¹⁷<BR>\n" );
document.write( "    * Raise equation (1) to the power of 7: (x⁵y¹⁷)⁷ = r⁷ => x³⁵y¹¹⁹ = r⁷<BR>\n" );
document.write( "    * Divide the second equation by the first: (x³⁵y¹¹⁹) / (x³⁴y¹¹⁹) = r⁷ / s¹⁷<BR>\n" );
document.write( "    * x = r⁷ / s¹⁷\r<BR>\n" );
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document.write( "* **Eliminate x:**<BR>\n" );
document.write( "    * Raise equation (2) to the power of 5: (x²y⁷)⁵ = s⁵ => x¹⁰y³⁵ = s⁵<BR>\n" );
document.write( "    * Raise equation (1) to the power of 2: (x⁵y¹⁷)² = r² => x¹⁰y³⁴ = r²<BR>\n" );
document.write( "    * Divide the first equation by the second: (x¹⁰y³⁵) / (x¹⁰y³⁴) = s⁵ / r²<BR>\n" );
document.write( "    * y = s⁵ / r²\r<BR>\n" );
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document.write( "**3. Compare with the Given Forms**\r<BR>\n" );
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document.write( "We have:<BR>\n" );
document.write( "* x = r⁷ / s¹⁷ = r⁷s⁻¹⁷<BR>\n" );
document.write( "* y = s⁵ / r² = s⁵r⁻²\r<BR>\n" );
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document.write( "Comparing with the given forms:<BR>\n" );
document.write( "* x = r^a / s^b = r^a s^-b<BR>\n" );
document.write( "* y = s^c / r^d = s^c r^-d\r<BR>\n" );
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document.write( "We can see:<BR>\n" );
document.write( "* a = 7<BR>\n" );
document.write( "* b = 17<BR>\n" );
document.write( "* c = 5<BR>\n" );
document.write( "* d = 2\r<BR>\n" );
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document.write( "**4. Calculate a + b + c + d**\r<BR>\n" );
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document.write( "* a + b + c + d = 7 + 17 + 5 + 2 = 31\r<BR>\n" );
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document.write( "**Therefore, a + b + c + d = 31**<BR>\n" );
document.write( " </SPAN>\n" );
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