Question 970671
<pre>
The other tutor only gave you an approximate decimal solution using
a calculator.  That's not what your teacher wants.  Your teacher wants
an exact answer.  Here's the way to do it using fraction exponents:

{{{root(3,12sqrt(12))}}}

Write both 12's as 2<sup>2</sup>·3

{{{root(3,2^2*3sqrt(2^2*3))}}}  

Write the square root as the {{{1/2}}} power
and the cube root as the {{{1/3}}} power:

{{{(2^2*3*(2^2*3)^(1/2))^(1/3)}}}

Make everything have an exponent by giving a 1 exponent
to whatever doesn't have an exponent showing:

{{{(2^2*3^1*(2^2*3^1)^(1/2))^(1/3)}}}

Remove the inner parentheses by multiplying exponents:

{{{(  2^2*3^1*2^  (2*expr(1/2)) *3^(1*expr(1/2))   )^(1/3)}}}

{{{(  2^2*3^1*2^1 *3^(1/2)   )^(1/3)}}}

Add the exponents of 2 and the exponents of 3

{{{(  2^3*3^(1&1/2)   )^(1/3)}}}

Change mixed number exponent {{{1&1/2}}} to {{{3/2}}}

{{{(  2^3*3^(3/2)   )^(1/3)}}}

Remove the remaining parentheses by multiplying exponents:

{{{matrix(2,1,"",

2^(3*expr(1/3))*3^(expr(3/2)*expr(1/3)))}}}

{{{matrix(2,1,"",

2^(cross(3)*expr(1/cross(3)))*3^(expr(cross(3)/2)*expr(1/cross(3))))}}}

{{{matrix(2,1,"",
2^1*3^(1/2))}}}

Erase the 1 exponent and change the {{{1/2}}} exponent to a square root

{{{2*sqrt(3)}}}

Edwin</pre>