How to rewrite $7-\sqrt 5$ in root form without a minus sign? How to rewrite $7-\sqrt 5$ in root form without a minus sign ? 
For clarity "root form " means an expression that only contains a finite amount of positive integers , additions , substractions , multiplications and root extractions (sqrt, cuberoot etc).
For example some quintic equations cannot be solved in root form.
A " root form without a minus sign " means an expression that only contains a finite amount of positive integers , additions , multiplications and root extractions (sqrt , cuberoot etc).
So the solution could look something like this :
$$ 7-\sqrt5 = \sqrt{...+1+(...)^{\frac{2}{3}}}+\sqrt{...+2(...)^{\frac{1}{3}}}$$
How to solve such problems ?
EDIT
Warning: $\dfrac{44}{7+\sqrt 5}$ is not a solution, since no divisions are allowed!
I got that answer 3 times now so I put it in the OP as a warning , not just the comments.

Let $n$ be some positive integer.
$(7-\sqrt 5)^n = r$ where $r$ is in the desired form but there is no root symbol over the entire expression on the RHS.
Now every root symbol in $r$ denotes a principal root.
The number of ways that we can write the same identity $r$ but replace one or more of its roots by a nonprincipal is given by the product $a_1 a_2 ... a_m = k$ where the amount of roots symbols are $m$ and the roots are $*^{1/a_i}$.
We denote the Original $r$ and the nonprincipal ones as 
$r_0 =r$
$r_i$
( i is just an index )
Therefore (by very basic galois theory) there exists an integer polynomial $P$ of degree $k$ such that $P(x) = (x-r_0)(x-r_1)(x-r_2)...(x-r_i)...(x-r_k).$
( notice that $P$ is completely factored and hence we have unique factorization )
Now clearly $r = r_0$ is the max in absolute value of all the $r_i$. 
Call this PROPERTY 1.
It is easy to show with some elementary ring theory or number theory that if
$(7-\sqrt 5)^n$ is a zero of an integer polynomial then so is $(7+\sqrt 5)^n$.
Therefore some $r_i$ must be equal to $(7+\sqrt 5)^n$.
This is PROPERTY 2.
Now notice that |$(7+\sqrt 5)^n$| > |$(7-\sqrt 5)^n$|.
This is PROPERTY 3.
Combining property 1,2 and 3 we see that $r$ must be both the largest in absolute value of all the $r_i$ and NOT the largest in absolute value of all the $r_i$.
Proof by contradiction.
QED.
Sorry if I did not use much of Galois theory , Im not so good with it.
