Question 1066053
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Evaluate a^3 + 1/a^3 if a + 1/a = 6.
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<pre>
Let  {{{a + 1/a}}} = 6.    (1)

Cube both sides of the equation (1). You will get

{{{a^3}}} + {{{3*a^2*(1/a)}}} + {{{3*a*(1/a^2)}}} + {{{1/a^3}}} = {{{6^3}}} =  216   (after using the formula for the cube of a sum {{{(x+y)^3}}} = {{{x^3 + 3*x^2*y + 3*x*y^2 + y^3}}}).

Now, cancel the factors {{{a}}} and {{{1/a}}}. You will get

{{{a^3}}} + {{{3*a}}} + {{{3*(1/a)}}} + {{{1/a^3}}} = {{{216}}},    or, after grouping the terms,

{{{a^3}}} + {{{3*(a + (1/a))}}} + {{{1/a^3}}} = {{{216}}}.

Replace {{{a + 1/a}}} by 6, in accordance with the condition. You will get

{{{a^3}}} + {{{3*6}}} + {{{1/a^3}}} = {{{216}}}    or

{{{a^3}}} + {{{1/a^3}}} = {{{216}}} - {{{3*6}}} = {{{216 - 18}}} = {{{198}}}.

Thus &nbsp;&nbsp;{{{a^3}}} + {{{1/a^3}}} = {{{198)}}}.

<U>Answer</U>. &nbsp;&nbsp;{{{a^3}}} + {{{1/a^3}}} = {{{198}}}. 
</pre>

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;{{{(x^3 + 1/x^3)}}}  &nbsp;and &nbsp;{{{(x^5+1/x^5)}}}</A>

in this site.


Also, 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 lesson is the part of this online textbook under the topic "<U>Evaluation, substitution</U>".