Question 1116707
If {{{a = (sqrt(3)-sqrt(2))/(sqrt(3)+sqrt(2))}}} and 
{{{b = (sqrt(3)+sqrt(2))/(sqrt(3)-sqrt(2))}}}
find the value of a²+b²-5ab
<pre>
We rationalize the denominator of a

{{{matrix(1,22,
a,
"",
""="",
(sqrt(3)-sqrt(2))/(sqrt(3)+sqrt(2)),
"",
""="",
"",
(sqrt(3)-sqrt(2))/(sqrt(3)+sqrt(2)),
"×",
(sqrt(3)-sqrt(2))/(sqrt(3)-sqrt(2)),
"",
""="",
"",
((sqrt(3)-sqrt(2))(sqrt(3)-sqrt(2)))/((sqrt(3)+sqrt(2))(sqrt(3)-sqrt(2))),
"",
""="",
"",
(3-2sqrt(6)+2)/(3-2),
"",
(5-2sqrt(6))/1,
"",
5-2sqrt(6)
)}}}

We rationalize the denominator of b

{{{matrix(1,22,
a,
"",
""="",
(sqrt(3)+sqrt(2))/(sqrt(3)-sqrt(2)),
"",
""="",
"",
(sqrt(3)+sqrt(2))/(sqrt(3)-sqrt(2)),
"×",
(sqrt(3)+sqrt(2))/(sqrt(3)+sqrt(2)),
"",
""="",
"",
((sqrt(3)+sqrt(2))(sqrt(3)+sqrt(2)))/((sqrt(3)-sqrt(2))(sqrt(3)+sqrt(2))),
"",
""="",
"",
(3+2sqrt(6)+2)/(3-2),
"",
(5+2sqrt(6))/1,
"",
5+2sqrt(6)
)}}}

We want to find {{{a^2+b^2-5ab}}}

{{{matrix(1,6,

a^2+b^2-5ab,
""="",
(5-2sqrt(6))^2+(5+2sqrt(6))^2-5(5-2sqrt(6))(5+2sqrt(6)),
""="",
25-20sqrt(6)+4*6+25+20sqrt(6)+4*6-5(25-4*6),
""="")}}}

{{{matrix(1,7,

25-20sqrt(6)+24+25+20sqrt(6)+24-5(25-24),
""="",
25-20sqrt(6)+24+25+20sqrt(6)+24-5(1),
""="",
25+24+25+24-5,
""="",
93)}}}

Edwin</pre>