Algebraic value | AIME I, 1990 | Question 2

Content

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Algebraic Value.

Algebraic value – AIME I, 1990

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.

Source

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.