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AMC 10 USA Math Olympiad

AMC 10A Year 2007 Problem 20 Sequential Hints

American Math Contest (AMC 10A) 2007, Problem 20 solution. Well, this is a very basic but interesting algebra problem where we can play with squares.

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Understand the problem

[/et_pb_text][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway||||||||” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]Suppose that the number \(a\) satisfies the equation \(a\)+\(\frac{1}{a}\)=\(4\) What is the value of \(a^4\)+\(\frac{1}{a^4}\)?  

(a) 164.        (b)172.       (c)192.       (d)194.       (e)212

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American Mathematical Contest 2007 10 A Problem 20

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Basic Algebra 

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4/10

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Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics 

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Start with hints

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Step 1. Look’s hard , lets see we are given with \(a\)+\(\frac{1}{a}\)=\(4\) and we need to find \(a^4\)+\(\frac{1}{a^4}\). So first we will square both sides in order to get  \(a^2\)+\(\frac{1}{a^2}\) and find its value. Give it a try!!!!.

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Step 2. We reached the power 2 now aim is to get to power 4. Now After getting the the value of  \(a^2\)+\(\frac{1}{a^2}\) , again square both sides to get the value of the expression \(a^4\)+\(\frac{1}{a^4}\) . Very close to the solution !!!!!!.

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Step 3 . Now we get the equation \(a^4\)+\(\frac{1}{a^4}\) after solving  \((a^2+\frac{1}{a^2})^2\). Now I hope u can see the answer by your self just a last step!!!!.

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Step 4 By solving we get the results that is   \(a^2\)+\(\frac{1}{a^2}\)=14

and \(a^4\)+\(\frac{1}{a^4}\) =194 which is our required value to find ,thats it!!!

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