The simplified value of `(sqrt(3)+1)(10+sqrt(12))(sqrt(12)-2)(5-sqrt(3))` is

Options

A. `16`
B. `88`
C. `176`
D. `132`

Answer

To simplify the expression \((\sqrt{3}+1)(10+\sqrt{12})(\sqrt{12}-2)(5-\sqrt{3})\), we will follow these steps:

### Step 1: Simplify \(\sqrt{12}\)
First, we simplify \(\sqrt{12}\):
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]
Thus, we can rewrite the expression:
\[ (\sqrt{3}+1)(10 + 2\sqrt{3})(2\sqrt{3} - 2)(5 - \sqrt{3}) \]

### Step 2: Factor out common terms
Notice that \(2\sqrt{3} - 2\) can be factored:
\[ 2(\sqrt{3} - 1) \]
Now the expression becomes:
\[ (\sqrt{3}+1)(10 + 2\sqrt{3}) \cdot 2(\sqrt{3} - 1)(5 - \sqrt{3}) \]

### Step 3: Combine the factors
Now we can combine the factors:
\[ 2(\sqrt{3}+1)(\sqrt{3}-1)(10 + 2\sqrt{3})(5 - \sqrt{3}) \]

### Step 4: Use the difference of squares
Using the identity \(a^2 - b^2 = (a+b)(a-b)\) on \((\sqrt{3}+1)(\sqrt{3}-1)\):
\[ (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2 \]
Thus, we have:
\[ 2 \cdot 2 \cdot (10 + 2\sqrt{3})(5 - \sqrt{3}) \]

### Step 5: Simplify further
Now we can simplify:
\[ 4(10 + 2\sqrt{3})(5 - \sqrt{3}) \]

### Step 6: Expand the product
Next, we expand \((10 + 2\sqrt{3})(5 - \sqrt{3})\):
\[ = 10 \cdot 5 - 10 \cdot \sqrt{3} + 2\sqrt{3} \cdot 5 - 2\sqrt{3} \cdot \sqrt{3} \]
\[ = 50 - 10\sqrt{3} + 10\sqrt{3} - 2 \cdot 3 \]
\[ = 50 - 6 = 44 \]

### Step 7: Final calculation
Now we multiply by 4:
\[ 4 \cdot 44 = 176 \]

Thus, the simplified value of the expression is:
\[ \boxed{176} \]