Question 1092216
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If a+1/a=7 find the value of (a-1/a)^2 and a^4+1/a^4
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<pre>
1.  {{{a+1/a}}} = 7  ====>  square both sides  ====>

    {{{a^2}}} + {{{2*a*(1/a)}}} + {{{1/a^2}}} = 49  ====>  notice that {{{a*(1/a)}}} = 1  ====>

    {{{a^2}}} + 2 + {{{1/a^2}}} = 49  ====>  {{{a^2}}} + {{{1/a^2}}} = 49 - 2  ====>  {{{a^2}}} + {{{1/a^2}}} = 47  ====>

    {{{a^2}}} - 2 + {{{1/a^2}}} = 47 - 2 = 45  ====>  {{{(a - 1/a)^2}}} = 45
</pre>

And the first part is solved.


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

    if  {{{a+1/a}}} = 7  then  {{{a^2}}} + {{{1/a^2}}} = 47.


    Square the last equality one more time. You will get

    {{{a^4}}} + {{{2*a^2*(1/a^2)}}} + {{{a^4}}} = {{{47^2}}}  ====>

    {{{a^4}}} + 2 + {{{1/a^4}}} = {{{47^2}}}  ====>

    {{{a^4}}} + {{{1/a^4}}} = {{{47^2-2}}} = 2207.
</pre>

And the second part is solved, too.


<U>Answer</U>.  &nbsp;If &nbsp;{{{a+1/a}}} = 7 &nbsp;then  &nbsp;{{{(a - 1/a)^2}}} = 45  &nbsp;and  &nbsp;{{{a^4}}} + {{{1/a^4}}} = 2207.



-------------------
See the lesson 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/HOW-TO-evaluate-expressions-involving-x%2Binv%28x%29-x2%2Binv%28x2%29-and-x%5E3%2Binv%28x%5E3%29.lesson>HOW TO evaluate expressions involving &nbsp;{{{(x + 1/x)}}}, &nbsp;{{{(x^2+1/x^2)}}} &nbsp;and &nbsp;{{{(x^3+1/x^3)}}}</A>

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


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



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.