If `sqrt(6)=2.55`, then the value of `sqrt((2)/(3))+3sqrt((3)/(2))` is

Options

A. `4.48`
B. `4.49`
C. `3.71`
D. None of these

Answer

To solve the expression \( \sqrt{\frac{2}{3}} + 3\sqrt{\frac{3}{2}} \) using the given value \( \sqrt{6} = 2.55 \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{\frac{2}{3}} + 3\sqrt{\frac{3}{2}} \]

### Step 2: Express in terms of \( \sqrt{6} \)
To use the value of \( \sqrt{6} \), we can rewrite \( \sqrt{\frac{2}{3}} \) and \( \sqrt{\frac{3}{2}} \):
\[ \sqrt{\frac{2}{3}} = \frac{\sqrt{2}}{\sqrt{3}} \quad \text{and} \quad \sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}} \]

### Step 3: Combine the terms
Now, we can express \( \sqrt{2} \) and \( \sqrt{3} \) in terms of \( \sqrt{6} \):
\[ \sqrt{2} = \frac{\sqrt{6}}{\sqrt{3}} \quad \text{and} \quad \sqrt{3} = \frac{\sqrt{6}}{\sqrt{2}} \]
Thus, we can rewrite the expression:
\[ \sqrt{\frac{2}{3}} = \frac{\sqrt{6}}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}} = \frac{\sqrt{6}}{3} \]
And for \( 3\sqrt{\frac{3}{2}} \):
\[ 3\sqrt{\frac{3}{2}} = 3 \cdot \frac{\sqrt{6}}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = \frac{3\sqrt{6}}{2} \]

### Step 4: Combine the fractions
Now we combine the two parts:
\[ \frac{\sqrt{6}}{3} + \frac{3\sqrt{6}}{2} \]
To add these fractions, we need a common denominator. The least common multiple of 3 and 2 is 6:
\[ \frac{2\sqrt{6}}{6} + \frac{9\sqrt{6}}{6} = \frac{(2 + 9)\sqrt{6}}{6} = \frac{11\sqrt{6}}{6} \]

### Step 5: Substitute the value of \( \sqrt{6} \)
Now, we substitute \( \sqrt{6} = 2.55 \):
\[ \frac{11 \cdot 2.55}{6} \]

### Step 6: Calculate the numerator
Calculating the numerator:
\[ 11 \cdot 2.55 = 28.05 \]

### Step 7: Divide by 6
Now we divide by 6:
\[ \frac{28.05}{6} = 4.675 \]

### Final Answer
Thus, the value of the expression \( \sqrt{\frac{2}{3}} + 3\sqrt{\frac{3}{2}} \) is:
\[ \boxed{4.675} \]