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# 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.

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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