document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209787>Question 1209787</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If 5^x = 7^y = 1225<BR>\n" );
document.write( "find xy/(x + y) <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850431 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=850431\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given that $5^x = 7^y = 1225$, we can use the following steps to find $\frac{xy}{x+y}$.\r<BR>\n" );
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document.write( "1. Express 1225 as a product of its prime factors.<BR>\n" );
document.write( "2. Substitute the prime factorization of 1225 into the given equations.<BR>\n" );
document.write( "3. Express x and y in terms of the exponents of the prime factors.<BR>\n" );
document.write( "4. Substitute the expressions for x and y into the expression $\frac{xy}{x+y}$.<BR>\n" );
document.write( "5. Simplify the expression.\r<BR>\n" );
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document.write( "Here are the detailed steps:\r<BR>\n" );
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document.write( "1. Express 1225 as a product of its prime factors.<BR>\n" );
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document.write( "   $1225 = 5^2 \cdot 7^2$\r<BR>\n" );
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document.write( "2. Substitute the prime factorization of 1225 into the given equations.\r<BR>\n" );
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document.write( "   $5^x = 5^2 \cdot 7^2$<BR>\n" );
document.write( "   $7^y = 5^2 \cdot 7^2$\r<BR>\n" );
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document.write( "3. Express x and y in terms of the exponents of the prime factors.\r<BR>\n" );
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document.write( "   $x = 2 + 2 \log_5 7$<BR>\n" );
document.write( "   $y = 2 + 2 \log_7 5$\r<BR>\n" );
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document.write( "4. Substitute the expressions for x and y into the expression $\frac{xy}{x+y}$.\r<BR>\n" );
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document.write( "   $\frac{xy}{x+y} = \frac{(2 + 2 \log_5 7)(2 + 2 \log_7 5)}{(2 + 2 \log_5 7) + (2 + 2 \log_7 5)}$\r<BR>\n" );
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document.write( "5. Simplify the expression.\r<BR>\n" );
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document.write( "   $\frac{xy}{x+y} = \frac{4 + 4 \log_5 7 + 4 \log_7 5 + 4 \log_5 7 \log_7 5}{4 + 2 \log_5 7 + 2 \log_7 5}$<BR>\n" );
document.write( "   <BR>\n" );
document.write( "   $\frac{xy}{x+y} = \frac{2 + 2 \log_5 7 + 2 \log_7 5 + 2 \log_5 7 \log_7 5}{2 + \log_5 7 + \log_7 5}$<BR>\n" );
document.write( "   <BR>\n" );
document.write( "   $\frac{xy}{x+y} = \frac{2 (1 + \log_5 7 + \log_7 5 + \log_5 7 \log_7 5)}{2 + \log_5 7 + \log_7 5}$<BR>\n" );
document.write( "   <BR>\n" );
document.write( "   $\frac{xy}{x+y} = \frac{2 (1 + \log_5 7)(1 + \log_7 5)}{2 + \log_5 7 + \log_7 5}$\r<BR>\n" );
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document.write( "   Since $\log_a b = \frac{1}{\log_b a}$, we have $\log_5 7 \log_7 5 = 1$.  Therefore,<BR>\n" );
document.write( "   \[\frac{xy}{x + y} = \frac{2 (1 + \log_5 7)(1 + \log_7 5)}{2 + \log_5 7 + \log_7 5} = \frac{2 (1 + \log_5 7 + \log_7 5 + 1)}{2 + \log_5 7 + \log_7 5} = \boxed{2}.\] </SPAN>\n" );
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