# Algebraic value | AIME I, 1990 | Question 2

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

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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