document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=672074>Question 672074</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How would you define what a rational exponent is? Is there such a thing as an irrational exponent? Explain.<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 #417802 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=417802\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Definition:\r<BR>\n" );
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document.write( "    If the power or the exponent raised on a number is in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Fq ALIGN=MIDDLE  ALT=\"p%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+%3C%3E0 ALIGN=MIDDLE  ALT=\"q+%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the number is said to have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=rational ALIGN=MIDDLE  ALT=\"rational\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+exponent ALIGN=MIDDLE  ALT=\"+exponent\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "For example: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"8%5E%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, means to take the 3-th root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Exponents can accept values from the multitude of the real numbers. They can be both rational or irational. \r<BR>\n" );
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document.write( "Irrational exponents: \r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > be an irrational number. Then, for a rational number  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Fn ALIGN=MIDDLE  ALT=\"m%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" > arbitrarily close <BR>\n" );
document.write( "to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can find a unique value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b+%3E+0 ALIGN=MIDDLE  ALT=\"+b+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > so that the rational exponent  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%5E%28m%2Fn%29 ALIGN=MIDDLE  ALT=\"+a%5E%28m%2Fn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes arbitrarily close to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We call such value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > the irrational exponent <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Ex ALIGN=MIDDLE  ALT=\"a%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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