Find the value of a and b respectively, if `(5+sqrt(3))/(7-4sqrt(3))=47a+sqrt(3)b` .

Options

A. `2,1`
B. `1,27`
C. `11,28`
D. `2,38`

Answer

To solve the equation \((5+\sqrt{3})/(7-4\sqrt{3})=47a+\sqrt{3}b\), we will follow these steps:

### Step 1: Rationalize the denominator
We start by rationalizing the denominator of the left-hand side. To do this, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(7 + 4\sqrt{3}\).

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

### Step 2: Simplify the denominator
Using the difference of squares formula, the denominator simplifies as follows:

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

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

\[ (5+\sqrt{3})(7+4\sqrt{3}) = 5 \cdot 7 + 5 \cdot 4\sqrt{3} + \sqrt{3} \cdot 7 + \sqrt{3} \cdot 4\sqrt{3} \]
\[ = 35 + 20\sqrt{3} + 7\sqrt{3} + 4 \cdot 3 \]
\[ = 35 + 20\sqrt{3} + 7\sqrt{3} + 12 \]
\[ = 47 + 27\sqrt{3} \]

### Step 4: Combine the results
Now, we can write the left-hand side as:

\[ \frac{47 + 27\sqrt{3}}{1} = 47 + 27\sqrt{3} \]

### Step 5: Compare with the right-hand side
We compare this with the right-hand side \(47a + \sqrt{3}b\):

\[ 47 + 27\sqrt{3} = 47a + \sqrt{3}b \]

### Step 6: Set up equations
From the comparison, we can set up the following equations:

1. \(47a = 47\)
2. \(b = 27\)

### Step 7: Solve for \(a\) and \(b\)
From the first equation:

\[ a = \frac{47}{47} = 1 \]

From the second equation:

\[ b = 27 \]

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

\[ a = 1, \quad b = 27 \]