If `(5+2sqrt3)/(7+4sqrt3)=a+bsqrt3`, then the value of a and b is

Options

A. a = 11, b = -6
B. a = 6, b = 11
C. a = -11, b = 6
D. a = -11, b = -6

Answer

To solve the equation \(\frac{5 + 2\sqrt{3}}{7 + 4\sqrt{3}} = a + b\sqrt{3}\), we will follow these steps:

### Step 1: Rationalize the Denominator
To eliminate the square root in the denominator, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \(7 + 4\sqrt{3}\) is \(7 - 4\sqrt{3}\).

\[ \frac{5 + 2\sqrt{3}}{7 + 4\sqrt{3}} \cdot \frac{7 - 4\sqrt{3}}{7 - 4\sqrt{3}} = \frac{(5 + 2\sqrt{3})(7 - 4\sqrt{3})}{(7 + 4\sqrt{3})(7 - 4\sqrt{3})} \]

### Step 2: Simplify the Denominator
Using the difference of squares formula \(a^2 - b^2\), we can simplify the denominator:

\[ (7 + 4\sqrt{3})(7 - 4\sqrt{3}) = 7^2 - (4\sqrt{3})^2 = 49 - 48 = 1 \]

### Step 3: Expand the Numerator
Now we expand the numerator:

\[ (5 + 2\sqrt{3})(7 - 4\sqrt{3}) = 5 \cdot 7 + 5 \cdot (-4\sqrt{3}) + 2\sqrt{3} \cdot 7 + 2\sqrt{3} \cdot (-4\sqrt{3}) \]

Calculating each term:

- \(5 \cdot 7 = 35\)
- \(5 \cdot (-4\sqrt{3}) = -20\sqrt{3}\)
- \(2\sqrt{3} \cdot 7 = 14\sqrt{3}\)
- \(2\sqrt{3} \cdot (-4\sqrt{3}) = -8 \cdot 3 = -24\)

Combining these gives:

\[ 35 - 20\sqrt{3} + 14\sqrt{3} - 24 = (35 - 24) + (-20\sqrt{3} + 14\sqrt{3}) = 11 - 6\sqrt{3} \]

### Step 4: Combine the Results
Now, we can write the entire expression:

\[ \frac{11 - 6\sqrt{3}}{1} = 11 - 6\sqrt{3} \]

### Step 5: Identify \(a\) and \(b\)
From the expression \(11 - 6\sqrt{3}\), we can equate it to \(a + b\sqrt{3}\):

\[ a = 11 \quad \text{and} \quad b = -6 \]

### Final Answer
Thus, the values of \(a\) and \(b\) are:

\[ \boxed{11} \quad \text{and} \quad \boxed{-6} \]