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