Let's work through the problem systematically:
Given:
[tex]\[
\sqrt{m} - \sqrt{n} = 2 \sqrt{2 - \sqrt{3}}
\][/tex]
### Step 1: Analyze [tex]\(2\sqrt{2-\sqrt{3}}\)[/tex]
We need to express [tex]\(2 \sqrt{2 - \sqrt{3}}\)[/tex] in a simpler form. Start by assuming:
[tex]\[
2 - \sqrt{3} = (\sqrt{a} - \sqrt{b})^2
\][/tex]
### Step 2: Expand and Compare
Expand the expression on the right-hand side:
[tex]\[
(\sqrt{a} - \sqrt{b})^2 = a + b - 2\sqrt{ab}
\][/tex]
Therefore:
[tex]\[
2 - \sqrt{3} = a + b - 2\sqrt{ab}
\][/tex]
### Step 3: Separate Real and Imaginary Parts
To satisfy the equation, equate the real and imaginary parts on both sides:
1. Real part:
[tex]\[
a + b = 2
\][/tex]
2. Imaginary part:
[tex]\[
-2\sqrt{ab} = -\sqrt{3} \Rightarrow 2\sqrt{ab} = \sqrt{3} \Rightarrow \sqrt{ab} = \frac{\sqrt{3}}{2}
\][/tex]
Square both sides:
[tex]\[
ab = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}
\][/tex]
### Step 4: Solve the System of Equations
Now we have the system:
[tex]\[
a + b = 2
\][/tex]
[tex]\[
ab = \frac{3}{4}
\][/tex]
These are the equations for which we solve for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]. The quadratic equation derived from these equations is:
[tex]\[
t^2 - (a + b)t + ab = 0 \Rightarrow t^2 - 2t + \frac{3}{4} = 0
\][/tex]
Solving this quadratic equation:
[tex]\[
t = \frac{2 \pm \sqrt{4 - 3}}{2} = \frac{2 \pm 1}{2}
\][/tex]
So, the solutions are:
[tex]\[
t = \frac{3}{2}, \quad t = \frac{1}{2}
\][/tex]
Thus, [tex]\(a = \frac{3}{2}\)[/tex] and [tex]\(b = \frac{1}{2}\)[/tex].
### Step 5: Substitute Back
Hence:
[tex]\[
\sqrt{2 - \sqrt{3}} = \sqrt{\frac{3}{2}} - \sqrt{\frac{1}{2}}
\][/tex]
And therefore:
[tex]\[
2\sqrt{2 - \sqrt{3}} = 2\left(\sqrt{\frac{3}{2}} - \sqrt{\frac{1}{2}}\right) = 2\left(\frac{\sqrt{6}}{2} - \frac{\sqrt{2}}{2}\right) = \sqrt{6} - \sqrt{2}
\][/tex]
### Step 6: Identify [tex]\(m\)[/tex] and [tex]\(n\)[/tex]
We need:
[tex]\[
\sqrt{m} - \sqrt{n} = \sqrt{6} - \sqrt{2}
\][/tex]
So, by comparing both sides:
[tex]\[
\sqrt{m} = \sqrt{6}, \quad \sqrt{n} = \sqrt{2}
\][/tex]
Squaring both sides, we find:
[tex]\[
m = 6, \quad n = 2
\][/tex]
Thus:
[tex]\[
\boxed{m = 6, \quad n = 2}
\][/tex]
