Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Algebraic Value.
Find the value of \((52+6\sqrt{43})^\frac{3}{2}-(52-6\sqrt{43})^\frac{3}{2}\).
Integers
Divisibility
Algebra
But try the problem first...
Answer: is 828.
AIME I, 1990, Question 2
Elementary Algebra by Hall and Knight
First hint
here we consider \(S^{2}=[(52+6\sqrt{43})^\frac{3}{2}-(52-6\sqrt{43})^\frac{3}{2}]^{2}\)
Second Hint
or, \(S^{2}=(52+6\sqrt{43})^{3}+(52-6\sqrt{43})^{3}\)
\(-2[(52+6\sqrt{43})(52-6\sqrt{43})]^\frac{3}{2}\)
Final Step
or, \(S^{2}\)=685584
or, S=828.
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