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 \((52+6\sqrt{43})^\frac{3}{2}-(52-6\sqrt{43})^\frac{3}{2}\).

  • is 107
  • is 828
  • is 840
  • cannot be determined from the given information

Key Concepts


Integers

Divisibility

Algebra

Check the Answer


But try the problem first…

Answer: is 828.

Source
Suggested Reading

AIME I, 1990, Question 2

Elementary Algebra by Hall and Knight

Try with Hints


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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