The expression `((5sqrt(3) + sqrt(50))(5-sqrt(24)))/(sqrt(75) - 5sqrt(2))` simplifies to

Options

A. 1
B. `sqrt(3) - sqrt(2)`
C. `sqrt(6) - sqrt(5)`
D. `sqrt(2)`

Answer

To simplify the expression \(\frac{(5\sqrt{3} + \sqrt{50})(5 - \sqrt{24})}{\sqrt{75} - 5\sqrt{2}}\), we will follow these steps:

### Step 1: Simplify \(\sqrt{50}\) and \(\sqrt{75}\)
- \(\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}\)
- \(\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}\)

### Step 2: Substitute back into the expression
Now we can rewrite the expression:
\[ \frac{(5\sqrt{3} + 5\sqrt{2})(5 - \sqrt{24})}{5\sqrt{3} - 5\sqrt{2}} \]

### Step 3: Factor out the common term
We can factor out \(5\) from the numerator and denominator:
\[ = \frac{5(\sqrt{3} + \sqrt{2})(5 - \sqrt{24})}{5(\sqrt{3} - \sqrt{2})} \]
This simplifies to:
\[ = \frac{(\sqrt{3} + \sqrt{2})(5 - \sqrt{24})}{\sqrt{3} - \sqrt{2}} \]

### Step 4: Rationalize the denominator
To rationalize the denominator, multiply the numerator and denominator by \((\sqrt{3} + \sqrt{2})\):
\[ = \frac{(\sqrt{3} + \sqrt{2})^2 (5 - \sqrt{24})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} \]

### Step 5: Simplify the denominator
Using the difference of squares:
\[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \]
So the denominator simplifies to \(1\).

### Step 6: Expand the numerator
Now we expand the numerator:
\[ (\sqrt{3} + \sqrt{2})^2 = (\sqrt{3})^2 + 2 \cdot \sqrt{3} \cdot \sqrt{2} + (\sqrt{2})^2 = 3 + 2\sqrt{6} + 2 = 5 + 2\sqrt{6} \]
Thus, the numerator becomes:
\[ (5 + 2\sqrt{6})(5 - \sqrt{24}) \]

### Step 7: Simplify \(\sqrt{24}\)
We know that \(\sqrt{24} = 2\sqrt{6}\), so:
\[ (5 + 2\sqrt{6})(5 - 2\sqrt{6}) \]

### Step 8: Use the difference of squares
Now apply the difference of squares:
\[ = 5^2 - (2\sqrt{6})^2 = 25 - 24 = 1 \]

### Final Result
Thus, the simplified value of the expression is:
\[ \boxed{1} \]