Express `1-(1)/(1+sqrt(3))+(1)/(1-sqrt(3))` in the form `a+bsqrt(3)`, where a and b are rational numbers, then the values of a and b respectively are

Options

A. `1,2`
B. `1,-1`
C. `3,1`
D. `2,1`

Answer

To express the expression \( 1 - \frac{1}{1+\sqrt{3}} + \frac{1}{1-\sqrt{3}} \) in the form \( a + b\sqrt{3} \), where \( a \) and \( b \) are rational numbers, we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ 1 - \frac{1}{1+\sqrt{3}} + \frac{1}{1-\sqrt{3}} \]

### Step 2: Find a common denominator
The denominators are \( 1+\sqrt{3} \) and \( 1-\sqrt{3} \). The common denominator is:
\[ (1+\sqrt{3})(1-\sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2 \]

### Step 3: Rewrite the fractions
Now we rewrite the fractions with the common denominator:
\[ 1 - \frac{(1-\sqrt{3})}{-2} + \frac{(1+\sqrt{3})}{-2} \]

### Step 4: Combine the fractions
Now combine the two fractions:
\[ 1 - \left( \frac{(1-\sqrt{3}) + (1+\sqrt{3})}{-2} \right) \]
This simplifies to:
\[ 1 - \frac{1 - \sqrt{3} + 1 + \sqrt{3}}{-2} = 1 - \frac{2}{-2} = 1 + 1 = 2 \]

### Step 5: Simplify the expression
Now we have:
\[ 2 + 0\sqrt{3} \]
This means \( a = 2 \) and \( b = 0 \).

### Final Result
Thus, the values of \( a \) and \( b \) are:
\[ a = 2, \quad b = 0 \]