Find the value of a and b in the following equation.
`(5+sqrt(3))/(7-4sqrt(3))=a+sqrt(3)b`

Options

A. `a=-27`, `b=47`
B. `a=-47`, `b=-27`
C. `a=47`, `b=27`
D. `a=27`, `b=47`

Answer

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

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

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

### Step 2: Calculate the Denominator
Using the difference of squares formula, we can simplify the denominator:

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

### Step 3: Calculate the Numerator
Now, we will expand the numerator:

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

Calculating each term:

- \(5 \cdot 7 = 35\)
- \(5 \cdot 4\sqrt{3} = 20\sqrt{3}\)
- \(\sqrt{3} \cdot 7 = 7\sqrt{3}\)
- \(\sqrt{3} \cdot 4\sqrt{3} = 4 \cdot 3 = 12\)

Now, combine the terms:

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

### Step 4: Combine the Results
Since the denominator is 1, we have:

\[ \frac{(5+\sqrt{3})(7+4\sqrt{3})}{1} = 47 + 27\sqrt{3} \]

### Step 5: Equate to the Right-Hand Side
Now we can equate this to the right-hand side \(a + \sqrt{3}b\):

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

From this, we can identify:

- \(a = 47\)
- \(b = 27\)

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

\[ a = 47, \quad b = 27 \]

---