Understand the problem

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

Source of the problem

American Mathematical Contest 2007 10 A Problem 20

Topic

Basic Algebra 

Difficulty Level

4/10

Suggested Book

Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics 

Start with hints

Do you really need a hint? Try it first!

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!!!!.

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 !!!!!!.

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!!!!.

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!!!