document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209710>Question 1209710</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 4^(1/√x) + 16^(1/√x) = 64^(1/√x)<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 #849938 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=849938\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this equation:\r<BR>\n" );
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document.write( "1. **Rewrite with a common base:** Notice that 4, 16, and 64 are all powers of 4:\r<BR>\n" );
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document.write( "*   4 = 4¹<BR>\n" );
document.write( "*   16 = 4²<BR>\n" );
document.write( "*   64 = 4³\r<BR>\n" );
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document.write( "So, the equation becomes:\r<BR>\n" );
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document.write( "4^(1/√x) + (4²)^(1/√x) = (4³)^(1/√x)\r<BR>\n" );
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document.write( "2. **Simplify using exponent rules:**\r<BR>\n" );
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document.write( "4^(1/√x) + 4^(2/√x) = 4^(3/√x)\r<BR>\n" );
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document.write( "3. **Substitute:** Let y = 4^(1/√x). Then the equation becomes:\r<BR>\n" );
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document.write( "y + y² = y³\r<BR>\n" );
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document.write( "4. **Rearrange:**\r<BR>\n" );
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document.write( "y³ - y² - y = 0\r<BR>\n" );
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document.write( "5. **Factor:**\r<BR>\n" );
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document.write( "y(y² - y - 1) = 0\r<BR>\n" );
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document.write( "6. **Solve for y:**\r<BR>\n" );
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document.write( "*   y = 0  (This is not possible since y = 4^(1/√x) and exponential functions are always positive.)\r<BR>\n" );
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document.write( "*   y² - y - 1 = 0\r<BR>\n" );
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document.write( "Use the quadratic formula to solve for y:\r<BR>\n" );
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document.write( "y = (1 ± √(1 + 4)) / 2<BR>\n" );
document.write( "y = (1 ± √5) / 2\r<BR>\n" );
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document.write( "Since y must be positive, we take the positive root:\r<BR>\n" );
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document.write( "y = (1 + √5) / 2  (This is the golden ratio, often represented by φ)\r<BR>\n" );
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document.write( "7. **Substitute back:** Now substitute y = 4^(1/√x) back into the equation:\r<BR>\n" );
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document.write( "4^(1/√x) = (1 + √5) / 2\r<BR>\n" );
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document.write( "8. **Take the logarithm of both sides (base 4 is convenient):**\r<BR>\n" );
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document.write( "log₄(4^(1/√x)) = log₄((1 + √5) / 2)\r<BR>\n" );
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document.write( "9. **Simplify:**\r<BR>\n" );
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document.write( "1/√x = log₄((1 + √5) / 2)\r<BR>\n" );
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document.write( "10. **Solve for x:**\r<BR>\n" );
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document.write( "√x = 1 / log₄((1 + √5) / 2)\r<BR>\n" );
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document.write( "x = 1 / [log₄((1 + √5) / 2)]²\r<BR>\n" );
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document.write( "11. **Change of base formula (optional):** It might be easier to calculate with natural logarithms (ln):\r<BR>\n" );
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document.write( "x = 1 / [(ln((1 + √5) / 2) / ln(4))]²<BR>\n" );
document.write( "x = 1 / [(ln((1 + √5) / 2) / 2ln(2))]²<BR>\n" );
document.write( "x = 4 / [ln((1 + √5) / 2)]²\r<BR>\n" );
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document.write( "12. **Calculate:** Using a calculator:\r<BR>\n" );
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document.write( "x ≈ 1.9999999... ≈ 2\r<BR>\n" );
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document.write( "Therefore, x = 2.<BR>\n" );
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