Question 1142422
<br>
------------------------------------------------<br>
NOTE: IGNORE THIS SOLUTION!<br>
The method was sound but the math was defective.  I absent-mindedly added 2 to both sides of the original equation instead of the required 4.<br>
See the response from tutor @ikleyn for the correct solution.<br>
--------------------------------
This is a common type of question on competitive math tests at the high school level.<br>
{{{4b^2+1/b^2 = 2}}}<br>
Note that both terms on the left are perfect squares:<br>
{{{(2b)^2 + (1/b)^2 = 2}}}<br>
Now remember that in squaring a binomial x+y, you get two perfect square terms plus a middle term: {{{(x+y)^2 = x^2+2xy+y^2}}}.<br>
Then note that if the binomial is of the form x+1/x, the middle term in the square is a constant: {{{(x+1/x)^2 = x^2+2+1/x^2}}}<br>
So, in this problem, add 2 to both sides; the left side becomes a perfect square trinomial:<br>
{{{(2b)^2 + 4 + (1/b)^2 = 4}}}
{{{(2b+1/b)^2 = 4}}}
{{{2b+1/b = 2}}}<br>
Now note that, in the expression you are to evaluate, both terms are perfect cubes: 8b^3 = (2b)^3; 1/b^3 = (1/b)^3.<br>
You will get both of those terms, plus other terms, when you cube (2b+1/b):<br>
{{{(2b+1/b)^3 = (2b)^3+3(2b)^2(1/b)+3(2b)(1/b)^2+(1/b)^3 = 8b^3+12b+6/b+1/b^3}}}<br>
This expression can be simplified using the value you now know of 2b+1/b:<br>
{{{8b^3+12b+6+1/b^3 = (8b^3+1/b^3)+6(2b+1/b) = (8b^3+1/b^3)+6(2) = (8b^3+1/b^3)+12}}}<br>
So now you have<br>
{{{(2b+1/b)^3 = 2^3 = 8 = (8b^3+1/b^3)+12}}}<br>
and so<br>
{{{8b^3+1/b^3 = 8-12 = -4}}}