# Difference between revisions of "2005 AIME II Problems/Problem 7"

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== Solution == | == Solution == | ||

+ | The expression for <math>x</math> looks very suspicious. We multiply top and bottom by <math>(\sqrt[16]{5} -1)</math>. | ||

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+ | <math>(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)(\sqrt[16]{5} -1) = (\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)((\sqrt[16]{5})^2 - 1) = (\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[8]{5} - 1) = </math> | ||

+ | <math>= (\sqrt{5}+1)(\sqrt[4]{5}+1)((\sqrt[8]{5})^2-1) = (\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[4]{5}-1) = (\sqrt{5}+1)((\sqrt[4]{5})^2-1) = (\sqrt5 + 1)(\sqrt5 - 1) = 4</math>. | ||

+ | (This is an example of a [[telescoping]] expression. An alternative way to recognize the telescoping nature would be to write roots as [[fractional exponent]]s, to write everything in terms of <math>\sqrt[16]5</math>, or to expand the denominator out entirely.) | ||

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+ | Thus, we have immediately that <math>x= \frac{4(\sqrt[16]5 - 1)}{4} = \sqrt[16]5 - 1</math> so <math>(x + 1)^{48} = (\sqrt[16]5)^{48} = 5^3 = 125</math> | ||

== See Also == | == See Also == | ||

*[[2005 AIME II Problems]] | *[[2005 AIME II Problems]] |

## Revision as of 09:12, 21 July 2006

## Problem

Let Find

## Solution

The expression for looks very suspicious. We multiply top and bottom by .

. (This is an example of a telescoping expression. An alternative way to recognize the telescoping nature would be to write roots as fractional exponents, to write everything in terms of , or to expand the denominator out entirely.)

Thus, we have immediately that so