document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209741>Question 1209741</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If 1/9^(1/x) + 1/3^(1/x) = 30, <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 #850064 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=850064\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve the equation 1/9^(1/x) + 1/3^(1/x) = 30:\r<BR>\n" );
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document.write( "1. **Rewrite with a common base:** Notice that 9 = 3². We can rewrite the first term as:\r<BR>\n" );
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document.write( "   1/9^(1/x) = 1/(3²)^(1/x) = 1/3^(2/x) = (1/3^(1/x))²\r<BR>\n" );
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document.write( "2. **Substitute:** Let y = 1/3^(1/x).  The equation becomes:\r<BR>\n" );
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document.write( "   y² + y = 30\r<BR>\n" );
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document.write( "3. **Rearrange:**\r<BR>\n" );
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document.write( "   y² + y - 30 = 0\r<BR>\n" );
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document.write( "4. **Factor:**\r<BR>\n" );
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document.write( "   (y + 6)(y - 5) = 0\r<BR>\n" );
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document.write( "5. **Solve for y:**\r<BR>\n" );
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document.write( "   y = -6 or y = 5\r<BR>\n" );
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document.write( "6. **Consider the valid solution:** Since y = 1/3^(1/x), y must be positive. Therefore, y = -6 is not a valid solution.  We are left with:\r<BR>\n" );
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document.write( "   y = 5\r<BR>\n" );
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document.write( "7. **Substitute back:**\r<BR>\n" );
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document.write( "   1/3^(1/x) = 5\r<BR>\n" );
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document.write( "8. **Rewrite:**\r<BR>\n" );
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document.write( "   3^(-1/x) = 5\r<BR>\n" );
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document.write( "9. **Take the logarithm of both sides (any base will work, but natural log is common):**\r<BR>\n" );
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document.write( "   ln(3^(-1/x)) = ln(5)\r<BR>\n" );
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document.write( "10. **Use logarithm power rule:**\r<BR>\n" );
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document.write( "   (-1/x) * ln(3) = ln(5)\r<BR>\n" );
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document.write( "11. **Solve for x:**\r<BR>\n" );
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document.write( "   -1/x = ln(5) / ln(3)<BR>\n" );
document.write( "   1/x = -ln(5) / ln(3)<BR>\n" );
document.write( "   x = -ln(3) / ln(5)\r<BR>\n" );
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document.write( "12. **Calculate:**\r<BR>\n" );
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document.write( "   x ≈ -1.465 / 1.609<BR>\n" );
document.write( "   x ≈ -0.911\r<BR>\n" );
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document.write( "Therefore, x ≈ -0.911.<BR>\n" );
document.write( " </SPAN>\n" );
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