document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202968>Question 1202968</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Evaluate the three radicals. Though they look similar, each one is different.<BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2816%2F49%29 ALIGN=MIDDLE  ALT=\"sqrt%2816%2F49%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "- <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2816%2F49%29 ALIGN=MIDDLE  ALT=\"sqrt%2816%2F49%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-16%2F49%29 ALIGN=MIDDLE  ALT=\"sqrt%28-16%2F49%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-16%2F49%29 ALIGN=MIDDLE  ALT=\"sqrt%28-16%2F49%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Which of the above square roots can be simplified with a real number answer and what is the simplified value?\r<BR>\n" );
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document.write( "Which of the above square roots cannot be simplified  and why can they  not be simplified?<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 #838185 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=838185\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2816%2F49%29+=+sqrt%28%284%5E2%29%2F%287%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%2816%2F49%29+=+sqrt%28%284%5E2%29%2F%287%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2816%2F49%29+=+%28sqrt%284%5E2%29%29%2F%28sqrt%287%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%2816%2F49%29+=+%28sqrt%284%5E2%29%29%2F%28sqrt%287%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2816%2F49%29+=+4%2F7 ALIGN=MIDDLE  ALT=\"sqrt%2816%2F49%29+=+4%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > The square root and squaring exponents cancel out\r<BR>\n" );
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document.write( "Through similar steps,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%2816%2F49%29+=+-4%2F7 ALIGN=MIDDLE  ALT=\"-sqrt%2816%2F49%29+=+-4%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Therefore, the first two items can be simplified to a real number result.\r<BR>\n" );
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document.write( "----------------------------\r<BR>\n" );
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document.write( "The last two items cannot be simplified to a real number result because of the negative under the square root.\r<BR>\n" );
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document.write( "But if we involve the imaginary number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i+=+sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"i+=+sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then we can say<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-16%2F49%29+=+%284%2F7%29i ALIGN=MIDDLE  ALT=\"sqrt%28-16%2F49%29+=+%284%2F7%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%28-16%2F49%29+=+-%284%2F7%29i ALIGN=MIDDLE  ALT=\"-sqrt%28-16%2F49%29+=+-%284%2F7%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Both of these results are complex numbers of the form a+bi\r<BR>\n" );
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document.write( "The reason a negative number is not allowed under the square root is to note how <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > produces nonnegative values when x is a real number.\r<BR>\n" );
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document.write( "Examples:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3 ALIGN=MIDDLE  ALT=\"x+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > leads to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+3%5E2+=+3%2A3+=+9 ALIGN=MIDDLE  ALT=\"x%5E2+=+3%5E2+=+3%2A3+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is nonnegative<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0 ALIGN=MIDDLE  ALT=\"x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > leads to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+0%5E2+=+0%2A0+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+=+0%5E2+=+0%2A0+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is nonnegative<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-5 ALIGN=MIDDLE  ALT=\"x+=+-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > leads to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+%28-5%29%5E2+=+%28-5%29%2A%28-5%29+=+25 ALIGN=MIDDLE  ALT=\"x%5E2+=+%28-5%29%5E2+=+%28-5%29%2A%28-5%29+=+25\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is nonnegative<BR>\n" );
document.write( "There is no way to have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > be negative when x is real.\r<BR>\n" );
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document.write( "So if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+y ALIGN=MIDDLE  ALT=\"x%5E2+=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" > then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sqrt%28y%29 ALIGN=MIDDLE  ALT=\"x+=+sqrt%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > shows that we can only plug in nonnegative values for y.<BR>\n" );
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