If `sqrt(3)=1.732`, then what is the value of `(sqrt(3)-(10)/(sqrt(3))+sqrt(27))` ?

Options

A. `1.154`
B. `1.577`
C. `1.464`
D. `2.358`

Answer

To solve the expression \(\sqrt{3} - \frac{10}{\sqrt{3}} + \sqrt{27}\), we will follow these steps:

### Step 1: Substitute the value of \(\sqrt{3}\)
Given that \(\sqrt{3} = 1.732\), we can substitute this value into the expression.

### Step 2: Simplify \(\sqrt{27}\)
We know that \(\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}\).

### Step 3: Rewrite the expression
Now, we can rewrite the expression as:
\[ \sqrt{3} - \frac{10}{\sqrt{3}} + 3\sqrt{3} \]

### Step 4: Combine like terms
Combine \(\sqrt{3}\) and \(3\sqrt{3}\):
\[ \sqrt{3} + 3\sqrt{3} = 4\sqrt{3} \]
So the expression becomes:
\[ 4\sqrt{3} - \frac{10}{\sqrt{3}} \]

### Step 5: Rationalize \(\frac{10}{\sqrt{3}}\)
To rationalize \(\frac{10}{\sqrt{3}}\), multiply the numerator and the denominator by \(\sqrt{3}\):
\[ \frac{10}{\sqrt{3}} = \frac{10\sqrt{3}}{3} \]

### Step 6: Substitute back into the expression
Now, substitute this back into the expression:
\[ 4\sqrt{3} - \frac{10\sqrt{3}}{3} \]

### Step 7: Find a common denominator
The common denominator for the terms is 3. Rewrite \(4\sqrt{3}\) as \(\frac{12\sqrt{3}}{3}\):
\[ \frac{12\sqrt{3}}{3} - \frac{10\sqrt{3}}{3} = \frac{(12\sqrt{3} - 10\sqrt{3})}{3} = \frac{2\sqrt{3}}{3} \]

### Step 8: Substitute the value of \(\sqrt{3}\)
Now, substitute \(\sqrt{3} = 1.732\) into \(\frac{2\sqrt{3}}{3}\):
\[ \frac{2 \times 1.732}{3} = \frac{3.464}{3} \]

### Step 9: Perform the division
Now, divide \(3.464\) by \(3\):
\[ 3.464 \div 3 \approx 1.15467 \]

### Final Answer
Thus, the value of the expression \(\sqrt{3} - \frac{10}{\sqrt{3}} + \sqrt{27}\) is approximately \(1.15467\).

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