Simplify : `sqrt(3 (33)/(64)) + sqrt(9 (1)/(7)) xx 2 sqrt(3 (1)/(9))`

Options

A. `(45)/(256)`
B. `1 (17)/(28)`
C. `4 (3)/(8)`
D. `2 (3)/(16)`

Answer

To simplify the expression `sqrt(3 (33)/(64)) + sqrt(9 (1)/(7)) xx 2 sqrt(3 (1)/(9))`, we will follow these steps:

### Step 1: Rewrite the expression
First, we rewrite the expression in a more manageable form:
\[ \sqrt{\frac{3 \times 33}{64}} + \sqrt{\frac{9 \times 1}{7}} \times 2 \sqrt{\frac{3 \times 1}{9}} \]

### Step 2: Simplify the first square root
Calculate the numerator for the first square root:
\[ 3 \times 33 = 99 \]
So, we have:
\[ \sqrt{\frac{99}{64}} = \frac{\sqrt{99}}{8} \]

### Step 3: Simplify the second square root
Calculate the second square root:
\[ \sqrt{\frac{9 \times 1}{7}} = \frac{3}{\sqrt{7}} \]
Now, multiply this by \(2\sqrt{\frac{3 \times 1}{9}}\):
\[ \sqrt{\frac{3 \times 1}{9}} = \frac{\sqrt{3}}{3} \]
Thus,
\[ 2 \times \frac{3}{\sqrt{7}} \times \frac{\sqrt{3}}{3} = \frac{2\sqrt{3}}{\sqrt{7}} \]

### Step 4: Combine the terms
Now we combine the two parts:
\[ \frac{\sqrt{99}}{8} + \frac{2\sqrt{3}}{\sqrt{7}} \]

### Step 5: Find a common denominator
To add these fractions, we need a common denominator. The common denominator will be \(8\sqrt{7}\):
\[ \frac{\sqrt{99} \cdot \sqrt{7}}{8\sqrt{7}} + \frac{2\sqrt{3} \cdot 8}{8\sqrt{7}} = \frac{\sqrt{693} + 16\sqrt{3}}{8\sqrt{7}} \]

### Step 6: Final expression
Thus, the simplified expression is:
\[ \frac{\sqrt{693} + 16\sqrt{3}}{8\sqrt{7}} \]