Find the value of `3 sqrt(147)-sqrt((1)/(3))-sqrt((1)/(27))`

`3 sqrt(147)-sqrt((1)/(3))-sqrt((1)/(27))` का मान ज्ञात कीजिए।

Options

A. `(281 sqrt(3))/(9)`
B. `(181 sqrt(3))/(3)`
C. `(185sqrt(3))/(9)`
D. `(381 sqrt(3))/(9)`

Explanation

To find the value of the expression \( 3 \sqrt{147} - \sqrt{\frac{1}{3}} - \sqrt{\frac{1}{27}} \), we will break it down step by step.

### Step 1: Simplify \( \sqrt{147} \)

First, we simplify \( \sqrt{147} \):
\[ \sqrt{147} = \sqrt{49 \times 3} = \sqrt{49} \times \sqrt{3} = 7\sqrt{3} \]

### Step 2: Substitute back into the expression

Now substitute \( \sqrt{147} \) back into the expression:
\[ 3 \sqrt{147} = 3 \times 7\sqrt{3} = 21\sqrt{3} \]

So, the expression becomes:
\[ 21\sqrt{3} - \sqrt{\frac{1}{3}} - \sqrt{\frac{1}{27}} \]

### Step 3: Simplify \( \sqrt{\frac{1}{3}} \) and \( \sqrt{\frac{1}{27}} \)

Next, we simplify \( \sqrt{\frac{1}{3}} \):
\[ \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} \]

Now simplify \( \sqrt{\frac{1}{27}} \):
\[ \sqrt{\frac{1}{27}} = \sqrt{\frac{1}{3^3}} = \frac{1}{\sqrt{27}} = \frac{1}{3\sqrt{3}} \]

### Step 4: Rewrite the expression

Now we can rewrite the expression:
\[ 21\sqrt{3} - \frac{1}{\sqrt{3}} - \frac{1}{3\sqrt{3}} \]

### Step 5: Combine the terms

To combine the terms, we need a common denominator. The common denominator for \( \sqrt{3} \) and \( 3\sqrt{3} \) is \( 3\sqrt{3} \).

Rewriting each term with the common denominator:
- The first term becomes:
\[ 21\sqrt{3} = \frac{21\sqrt{3} \times 3}{3} = \frac{63\sqrt{3}}{3} \]
- The second term becomes:
\[ -\frac{1}{\sqrt{3}} = -\frac{1 \times 3}{3\sqrt{3}} = -\frac{3}{3\sqrt{3}} \]
- The third term remains:
\[ -\frac{1}{3\sqrt{3}} = -\frac{1}{3\sqrt{3}} \]

Now combine these:
\[ \frac{63\sqrt{3}}{3} - \frac{3}{3\sqrt{3}} - \frac{1}{3\sqrt{3}} = \frac{63\sqrt{3} - 3 - 1}{3\sqrt{3}} = \frac{63\sqrt{3} - 4}{3\sqrt{3}} \]

### Final Expression

Thus, the final value of the expression is:
\[ \frac{63\sqrt{3} - 4}{3\sqrt{3}} \]