If `(6 sqrt(3)+7 sqrt(2))/(sqrt(108)+ sqrt(50))=a+b sqrt(6)` then find the value of `(6a)/(b)`

Options

A. 9
B. 19
C. 38
D. 42

Answer

To solve the equation \(\frac{6\sqrt{3} + 7\sqrt{2}}{\sqrt{108} + \sqrt{50}} = a + b\sqrt{6}\), we will follow these steps:

### Step 1: Simplify the denominator
First, we simplify the square roots in the denominator:
\[ \sqrt{108} = \sqrt{36 \cdot 3} = 6\sqrt{3} \]
\[ \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2} \]
Thus, the denominator becomes:
\[ \sqrt{108} + \sqrt{50} = 6\sqrt{3} + 5\sqrt{2} \]

### Step 2: Rewrite the equation
Now we can rewrite the equation:
\[ \frac{6\sqrt{3} + 7\sqrt{2}}{6\sqrt{3} + 5\sqrt{2}} = a + b\sqrt{6} \]

### Step 3: Rationalize the denominator
To simplify the left side, we can multiply the numerator and denominator by the conjugate of the denominator:
\[ (6\sqrt{3} + 7\sqrt{2})(6\sqrt{3} - 5\sqrt{2}) \quad \text{and} \quad (6\sqrt{3} + 5\sqrt{2})(6\sqrt{3} - 5\sqrt{2}) \]

Calculating the denominator:
\[ (6\sqrt{3})^2 - (5\sqrt{2})^2 = 108 - 50 = 58 \]

Calculating the numerator:
\[ (6\sqrt{3})(6\sqrt{3}) - (6\sqrt{3})(5\sqrt{2}) + (7\sqrt{2})(6\sqrt{3}) - (7\sqrt{2})(5\sqrt{2}) \]
This simplifies to:
\[ 108 - 30\sqrt{6} + 42\sqrt{6} - 70 = 38 + 12\sqrt{6} \]

### Step 4: Combine results
Now we have:
\[ \frac{38 + 12\sqrt{6}}{58} = a + b\sqrt{6} \]

### Step 5: Split the fraction
This can be separated into:
\[ \frac{38}{58} + \frac{12\sqrt{6}}{58} = a + b\sqrt{6} \]
Thus:
\[ a = \frac{38}{58} = \frac{19}{29}, \quad b = \frac{12}{58} = \frac{6}{29} \]

### Step 6: Find \( \frac{6a}{b} \)
Now we need to find \( \frac{6a}{b} \):
\[ \frac{6a}{b} = \frac{6 \cdot \frac{19}{29}}{\frac{6}{29}} = \frac{6 \cdot 19}{6} = 19 \]

### Final Answer
Thus, the value of \( \frac{6a}{b} \) is \( \boxed{19} \).

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