document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=541277>Question 541277</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me How to solve this equation containing a rational exponent on the variable. \r<BR>\n" );
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document.write( "x 4/3= 81\r<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 #354162 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=354162\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You could use logarithms to work this problem, but I'm not sure that you have gotten that far along yet. So let's look at another way. <BR>\n" );
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document.write( "Recall the power rule of exponents. For example think of:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3%29%5E4 ALIGN=MIDDLE  ALT=\"%28x%5E3%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can raise <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the 4th power by multiplying the two exponents 4 and 3 to get 12. Another way of looking at it is that you get the 4th power of a quantity by multiplying that quantity by itself 4 times. So x-cubed multiplied by itself 4 times is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2Ax%5E3%2Ax%5E3%2Ax%5E3 ALIGN=MIDDLE  ALT=\"x%5E3%2Ax%5E3%2Ax%5E3%2Ax%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you find that result by adding the exponents to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2Ax%5E3%2Ax%5E3%2Ax%5E3+=+x%5E12 ALIGN=MIDDLE  ALT=\"x%5E3%2Ax%5E3%2Ax%5E3%2Ax%5E3+=+x%5E12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "A word of caution. Since an even power is involved, the base can either be positive or negative and you will get the same positive result because both a positive value and the same negative value will give the same positive result when they are raised to an even power.<BR>\n" );
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document.write( "That's just an example to show you that the power rule of multiplying exponents actually works. <BR>\n" );
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document.write( "The general form of the power rule is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5Ea%29%5Eb+=+x%5E%28a%2Ab%29 ALIGN=MIDDLE  ALT=\"%28x%5Ea%29%5Eb+=+x%5E%28a%2Ab%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now with that in mind, we can view <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%284%2F3%29 ALIGN=MIDDLE  ALT=\"x%5E%284%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > in a couple of ways. We can think of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F3 ALIGN=MIDDLE  ALT=\"4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > as being the product of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A%281%2F3%29 ALIGN=MIDDLE  ALT=\"4%2A%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So, using the power rule we can rewrite the left side of the problem equation as:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E4%29%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"%28x%5E4%29%5E%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E%281%2F3%29%29%5E4 ALIGN=MIDDLE  ALT=\"%28x%5E%281%2F3%29%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "either way will help us solve the problem. Recall that the exponent <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > means the cube root. <BR>\n" );
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document.write( "Now think about the right side of the equation problem. And note that 81 is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E4 ALIGN=MIDDLE  ALT=\"3%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So let's rewrite the entire equation as:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E%281%2F3%29%29%5E4=3%5E4 ALIGN=MIDDLE  ALT=\"%28x%5E%281%2F3%29%29%5E4=3%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now if we take the 4th root of both sides we get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2F3%29+=+3 ALIGN=MIDDLE  ALT=\"x%5E%281%2F3%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and since the 4th root is \"even\" not \"odd\", the result can be either positive or negative.<BR>\n" );
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document.write( "And we can read this as \"the cube root of x equals either +3 or -3.\" This can be inverted to: What are the numbers that if you take the cube root of them, the answer is either +3 or -3? And this can be further translated also to the answers: +3 cubed is what number and -3 cubed is what number? The answers are <BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+3%5E3 ALIGN=MIDDLE  ALT=\"+x+=+3%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and this is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+27 ALIGN=MIDDLE  ALT=\"x+=+27\"   DISABLED_style= \"max-width:90%; height:auto;\" > or<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+%28-3%29%5E3 ALIGN=MIDDLE  ALT=\"+x+=+%28-3%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and this is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-27 ALIGN=MIDDLE  ALT=\"x+=+-27\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now check by back-tracking. If we take the cube root of 27, we get both +3 and -3, and then we raise these two numbers to the 4th power and we get the same answer ... +81.<BR>\n" );
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document.write( "Or we could take both +27 and -27 and raise them to the 4th power and take the cube root of that to see if we get +81. Replace either +27 or -27 with their equivalent values of 3*3*3 (which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E3 ALIGN=MIDDLE  ALT=\"3%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >) or (-3)*(-3)*(-3) (which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3%29%5E3 ALIGN=MIDDLE  ALT=\"%28-3%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >). Raise these two quantities to the 4th power and you have the positive value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E12 ALIGN=MIDDLE  ALT=\"3%5E12\"   DISABLED_style= \"max-width:90%; height:auto;\" >. And finally take the cube root of that positive quantity by dividing its exponent by 3 to get the positive value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E4=81 ALIGN=MIDDLE  ALT=\"3%5E4=81\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "Either way it checks out. The answers to this problem are that x can be either +27 or -27.<BR>\n" );
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document.write( "Hope that as you think your way through this problem you understand a little more about a way that you can work with rational exponents.<BR>\n" );
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