SOLUTION: If a+1/a=7 find the value of (a-1/a)^2 and a^4+1/a^4

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Question 1092216: If a+1/a=7 find the value of
(a-1/a)^2 and a^4+1/a^4
Found 3 solutions by MathLover1, ikleyn, MathTherapy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

If %28a%2B1%29%2Fa=7->a%2B1=7a->1=7a-a->1=6a->a=1%2F6

then, the value of

and

Answer by ikleyn(52884) About Me  (Show Source):
You can put this solution on YOUR website!
.
If a+1/a=7 find the value of (a-1/a)^2 and a^4+1/a^4
~~~~~~~~~~~~~~~~ 

1.  a%2B1%2Fa = 7  ====>  square both sides  ====>

    a%5E2 + 2%2Aa%2A%281%2Fa%29 + 1%2Fa%5E2 = 49  ====>  notice that a%2A%281%2Fa%29 = 1  ====>

    a%5E2 + 2 + 1%2Fa%5E2 = 49  ====>  a%5E2 + 1%2Fa%5E2 = 49 - 2  ====>  a%5E2 + 1%2Fa%5E2 = 47  ====>

    a%5E2 - 2 + 1%2Fa%5E2 = 47 - 2 = 45  ====>  %28a+-+1%2Fa%29%5E2 = 45

And the first part is solved. 

2.  Notice that in the part "1" we just deduced as an intermediate result that

    if  a%2B1%2Fa = 7  then  a%5E2 + 1%2Fa%5E2 = 47.


    Square the last equality one more time. You will get

    a%5E4 + 2%2Aa%5E2%2A%281%2Fa%5E2%29 + a%5E4 = 47%5E2  ====>

    a%5E4 + 2 + 1%2Fa%5E4 = 47%5E2  ====>

    a%5E4 + 1%2Fa%5E4 = 47%5E2-2 = 2207.

And the second part is solved, too.

Answer.  If  a%2B1%2Fa = 7  then  %28a+-+1%2Fa%29%5E2 = 45  and  a%5E4 + 1%2Fa%5E4 = 2207.


-------------------
See the lesson
    - HOW TO evaluate expressions involving  %28x+%2B+1%2Fx%29,  %28x%5E2%2B1%2Fx%5E2%29  and  %28x%5E3%2B1%2Fx%5E3%29
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Evaluation, substitution".


Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.



Answer by MathTherapy(10556) About Me  (Show Source):
You can put this solution on YOUR website!

If a+1/a=7 find the value of
(a-1/a)^2 and a^4+1/a^4
a+%2B+1%2Fa+=+7
%28a+%2B+1%2Fa%29%5E2+=+7%5E2 ------ Squaring both sides
a%5E2+%2B+1+%2B+1+%2B+1%2Fa%5E2+=+49 
a%5E2+%2B+1%2Fa%5E2+=+47 ------- eq (i)


 -------- Substituting 47 for a%5E2+%2B+1%2Fa%5E2


a%5E2+%2B+1%2Fa%5E2+=+47
%28a%5E2+%2B+1%2Fa%5E2%29%5E2+=+47%5E2 --------- Squaring eq (i)
a%5E4+%2B+1+%2B+1+%2B+1%2Fa%5E4+=+47%5E2 
a%5E4+%2B+1%2Fa%5E4+%2B+2+=+47%5E2