Simplest form of `((sqrt(26 - 15sqrt(3)))/(5sqrt2-sqrt(38+5sqrt3)))^2` is

Options

A. 3
B. 9
C. `1/9`
D. `1/3`

Answer

To simplify the expression \(\left(\frac{\sqrt{26 - 15\sqrt{3}}}{5\sqrt{2} - \sqrt{38 + 5\sqrt{3}}}\right)^2\), we will follow these steps:

### Step 1: Simplify the Numerator
We start with the numerator \(\sqrt{26 - 15\sqrt{3}}\). We can express \(26 - 15\sqrt{3}\) in a different form.

Assume \(26 - 15\sqrt{3} = (\sqrt{a} - \sqrt{b})^2\).
Expanding this gives us:
\[ a + b - 2\sqrt{ab} = 26 - 15\sqrt{3} \]
From this, we can set up the equations:
1. \(a + b = 26\)
2. \(-2\sqrt{ab} = -15\sqrt{3}\) → \(2\sqrt{ab} = 15\sqrt{3}\) → \(\sqrt{ab} = \frac{15\sqrt{3}}{2}\) → \(ab = \left(\frac{15\sqrt{3}}{2}\right)^2 = \frac{225 \cdot 3}{4} = \frac{675}{4}\)

Now we need to solve the system of equations:
\[ a + b = 26 \]
\[ ab = \frac{675}{4} \]

### Step 2: Solve for \(a\) and \(b\)
Let \(a\) and \(b\) be the roots of the quadratic equation \(x^2 - (a+b)x + ab = 0\):
\[ x^2 - 26x + \frac{675}{4} = 0 \]
Multiplying through by 4 to eliminate the fraction:
\[ 4x^2 - 104x + 675 = 0 \]
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ x = \frac{104 \pm \sqrt{(-104)^2 - 4 \cdot 4 \cdot 675}}{2 \cdot 4} \]
Calculating the discriminant:
\[ 104^2 = 10816 \]
\[ 4 \cdot 4 \cdot 675 = 10800 \]
\[ b^2 - 4ac = 10816 - 10800 = 16 \]
Thus:
\[ x = \frac{104 \pm 4}{8} = \frac{108}{8} = 13.5 \quad \text{or} \quad \frac{100}{8} = 12.5 \]
So, \(a = 13.5\) and \(b = 12.5\) or vice versa.

### Step 3: Rewrite the Numerator
Thus, we can write:
\[ \sqrt{26 - 15\sqrt{3}} = \sqrt{(\sqrt{13.5} - \sqrt{12.5})^2} = \sqrt{13.5} - \sqrt{12.5} \]

### Step 4: Simplify the Denominator
Next, we simplify the denominator \(5\sqrt{2} - \sqrt{38 + 5\sqrt{3}}\).

Assume \(38 + 5\sqrt{3} = (\sqrt{c} + \sqrt{d})^2\).
Expanding gives:
\[ c + d + 2\sqrt{cd} = 38 + 5\sqrt{3} \]
Setting up the equations:
1. \(c + d = 38\)
2. \(2\sqrt{cd} = 5\sqrt{3}\) → \(\sqrt{cd} = \frac{5\sqrt{3}}{2}\) → \(cd = \left(\frac{5\sqrt{3}}{2}\right)^2 = \frac{75}{4}\)

### Step 5: Solve for \(c\) and \(d\)
Using the same method as before, we can find \(c\) and \(d\):
\[ x^2 - 38x + \frac{75}{4} = 0 \]
Multiplying through by 4:
\[ 4x^2 - 152x + 75 = 0 \]
Calculating the discriminant:
\[ (-152)^2 - 4 \cdot 4 \cdot 75 = 23104 - 1200 = 21904 \]
Thus:
\[ x = \frac{152 \pm \sqrt{21904}}{8} \]
Calculating \(\sqrt{21904} = 148\):
\[ x = \frac{152 \pm 148}{8} \]
This gives us roots \(x = 37.5\) and \(x = 0.5\).

### Step 6: Rewrite the Denominator
Thus:
\[ \sqrt{38 + 5\sqrt{3}} = \sqrt{(\sqrt{37.5} + \sqrt{0.5})^2} = \sqrt{37.5} + \sqrt{0.5} \]

### Step 7: Combine the Results
Now we can rewrite the original expression:
\[ \frac{\sqrt{13.5} - \sqrt{12.5}}{5\sqrt{2} - (\sqrt{37.5} + \sqrt{0.5})} \]

### Step 8: Square the Result
Finally, we square the entire expression:
\[ \left(\frac{\sqrt{13.5} - \sqrt{12.5}}{5\sqrt{2} - (\sqrt{37.5} + \sqrt{0.5})}\right)^2 \]

### Final Result
After simplification, we find:
\[ \left(\frac{\sqrt{13.5} - \sqrt{12.5}}{5\sqrt{2} - (\sqrt{37.5} + \sqrt{0.5})}\right)^2 = \frac{1}{3} \]

Thus, the simplest form of the expression is \(\frac{1}{3}\).