document.write( "Question 1171178: The radical √(85+√2736) can be expressed as √a + √b, where a < b. What is the product of ab?\r
\n" ); document.write( "\n" ); document.write( "(I know I'm supposed to solve this using a^2 + 2ab + b^2 = (a+b)^2, but have no idea as to how to implement it. Any advice/tips/help would be greatly appreciated) \n" ); document.write( "

Algebra.Com's Answer #796096 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "The radical √(85+√2736) can be expressed as √a + √b, where a < b. What is the product of ab?\r
\n" ); document.write( "\n" ); document.write( "(I know I'm supposed to solve this using a^2 + 2ab + b^2 = (a+b)^2, but have no idea as to how to implement it.
\n" ); document.write( "Any advice/tips/help would be greatly appreciated)
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\n" ); document.write( "\n" ); document.write( "I think   (no, I am 129% sure)   that in this form, the problem \r
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\n" ); document.write( "\n" ); document.write( "has INFINITELY MANY solutions, which is not a case you are looking/seeking for.\r
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\n" ); document.write( "\n" ); document.write( "To have a UNIQUE solution, the ADDITIONAL condition MUST be imposed: \r
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\n" ); document.write( "\n" ); document.write( "            the numbers \"a\" and \"b\" MUST be integer.\r
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\n" ); document.write( "\n" ); document.write( "When you REALIZE the necessity of this condition, you may RE-POST your problem to the forum, correctly formulated.\r
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\n" ); document.write( "\n" ); document.write( "Please DO NOT post it to me personally.\r
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\n" ); document.write( "\n" ); document.write( "Hey, and I have a question:\r
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\n" ); document.write( "\n" ); document.write( "        From which source do you retrieve the problems, formulated incorrectly ?\r
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\n" ); document.write( "\n" ); document.write( "To tutor @MathTutor_2020\r
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\n" ); document.write( "\n" ); document.write( "It is with regret I see that you missed or ignored the notice in my post,
\n" ); document.write( "saying that this problem REQUIRES an additional assumption that the numbers \"a\" and \"b\" are integer.\r
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\n" ); document.write( "\n" ); document.write( "Without this assumption, the problem has INFINITELY MANY solutions.\r
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\n" ); document.write( "\n" ); document.write( "I want to point some logical holes in your solution.\r
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\n" ); document.write( "\n" ); document.write( "First hole is THIS:\r
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document.write( "    In the main stream of your solution, you make an assumption that a + b = 85; \r\n" );
document.write( "    it allows you to conclude that  =  and then that ab = 684.\r\n" );
document.write( "    But, in order for this logical chain be consistent, you MUST CHECK that it is compatible with a+b = 85.\r\n" );
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document.write( "    For it, you actually MUST to determine both \"a\" and \"b\" separately,\r\n" );
document.write( "    which you claim \"unnecessary\" at the end of your solution.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Second hole is THIS :\r
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document.write( "    After squaring the original equation, you come to THIS equation\r\n" );
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document.write( "        = .\r\n" );
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document.write( "    This equation is in TWO VARIABLES, and, THEREFORE, has INFINITELY MANY solutions, \r\n" );
document.write( "    as it is clear to ANYONE with the university mathematical education.\r\n" );
document.write( "    Therefore, the assumption that \"a\" and \"b\" are integer, is NECESSARY.\r\n" );
document.write( "    It was the MAJOR POINT which I noted in my post above.\r\n" );
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\n" ); document.write( "\n" ); document.write( "I ask you PLEASE be more attentive to my posts in the future.\r
\n" ); document.write( "\n" ); document.write( "Otherwise, I will have this unpleasant duty to explain you simple things.\r
\n" ); document.write( "\n" ); document.write( "It does not mean that I am intolerant to criticism. In opposite, if you point my error, I will be happy to accept your notice.\r
\n" ); document.write( "\n" ); document.write( "But if I state right things, I don't like when people ignore it.\r
\n" ); document.write( "\n" ); document.write( "In simple words, I don't like to be in FALSE position (as, probably, any other person).\r
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\n" ); document.write( "\n" ); document.write( "Couple words about myself.\r
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document.write( "    I have PhD degree in Physics and Mathematics.\r\n" );
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document.write( "    I graduated from the Mathematical department of the University, which was ranked (at the time of my studies there)\r\n" );
document.write( "    higher than Princeton, MIT, Yell, Harvard, Caltech - higher than any US university at that time.\r\n" );
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document.write( "    Our teachers trained us to many things in Math, and, in particular, to see ANY (even smallest) holes in mathematical \r\n" );
document.write( "    texts, so I have an absolute hearing on it (similar to as other people have absolute hearing of music).\r\n" );
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\n" ); document.write( "\n" ); document.write( "I attentively looked in your posts at this forum, and I like your style and contribution.\r
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\n" ); document.write( "\n" ); document.write( "Best regards.\r
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\n" ); document.write( "\n" ); document.write( "@ikleyn\r
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