How do I prove that \(\displaystyle\sqrt{{{3}}}{\left\lbrace{5}\sqrt{{{2}}}+{7}\right\rbrace}-\sqrt{{{3}}}{\left\lbrace{5}\sqrt{{{2}}}-{7}\right\rbrace}\) is

Henry Winters 2022-04-12 Answered

How do I prove that 52+73-52-73 is an integer?

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Answers (2)

mortaraskxnl
Answered 2022-04-13 Author has 12 answers

If we numerically calculate 52+73 it looks an awful lot like 1+2 which is a big hint. Let’s prove that first:
(1+2)3=1+32+3(2)2+(2)3=7+52
Similarly,
(1+2)3=1+323(2)2+(2)3=7+52
Now that we have those cube roots in recognizable form, the rest is easy:
52+73-52-73=(1+2)-(-1+2)=2

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Kendall Wilkinson
Answered 2022-04-14 Author has 17 answers

x=52+73-52-73
x3=14-3(50-493)x
x3=143x
x3+3x14=0
(x2)(x2+2x+7)=0
Since the discriminant of the second bracket is less than 0, we can ignore it, and continue with
x2=0
x=2

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By Euler's formula, I can express i in the following way:
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I wonder if it is legitimate to write
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