When `(4+sqrt(7))` is presented in the form of perfect square it will be equal to

Options

A. `(2+sqrt(7))^(2)`
B. `(sqrt(7)/(2)+(1)/(2))^(2)`
C. `{(1)/(sqrt(2))(sqrt(7)+1)}^(2)`
D. `(sqrt(3)+sqrt(4))^(2)`

Answer

To express \(4 + \sqrt{7}\) in the form of a perfect square, we can follow these steps:

### Step 1: Identify the perfect square form
A perfect square can generally be expressed in the form \((a + b)^2\), where \(a\) and \(b\) are real numbers. We want to find \(a\) and \(b\) such that:

\[ (a + b)^2 = a^2 + b^2 + 2ab \]

### Step 2: Set up the equation
We want to find \(a\) and \(b\) such that:

\[ a^2 + b^2 + 2ab = 4 + \sqrt{7} \]

### Step 3: Choose values for \(a\) and \(b\)
Let's try \(a = 2\) and \(b = \frac{\sqrt{7}}{2}\). We will check if this choice satisfies the equation.

### Step 4: Calculate \(a^2\), \(b^2\), and \(2ab\)
1. Calculate \(a^2\):
\[ a^2 = 2^2 = 4 \]

2. Calculate \(b^2\):
\[ b^2 = \left(\frac{\sqrt{7}}{2}\right)^2 = \frac{7}{4} \]

3. Calculate \(2ab\):
\[ 2ab = 2 \cdot 2 \cdot \frac{\sqrt{7}}{2} = 2\sqrt{7} \]

### Step 5: Combine the results
Now, we can combine these results:

\[ a^2 + b^2 + 2ab = 4 + \frac{7}{4} + 2\sqrt{7} \]

To combine \(4\) and \(\frac{7}{4}\), convert \(4\) into quarters:

\[ 4 = \frac{16}{4} \]

Thus,

\[ a^2 + b^2 + 2ab = \frac{16}{4} + \frac{7}{4} + 2\sqrt{7} = \frac{23}{4} + 2\sqrt{7} \]

### Step 6: Check if it equals \(4 + \sqrt{7}\)
We need to check if:

\[ \frac{23}{4} + 2\sqrt{7} = 4 + \sqrt{7} \]

Convert \(4\) into quarters:

\[ 4 = \frac{16}{4} \]

So we compare:

\[ \frac{23}{4} + 2\sqrt{7} \quad \text{and} \quad \frac{16}{4} + \sqrt{7} \]

### Step 7: Simplify
The equation simplifies to:

\[ \frac{23}{4} + 2\sqrt{7} \neq \frac{16}{4} + \sqrt{7} \]

### Conclusion
After checking various options, we find that:

\[ 4 + \sqrt{7} = \left(\frac{1}{\sqrt{2}}(1 + \sqrt{7})\right)^2 \]

Thus, the expression \(4 + \sqrt{7}\) can be represented as a perfect square.

### Final Answer
The correct representation of \(4 + \sqrt{7}\) in the form of a perfect square is:

\[ \left(\frac{1}{\sqrt{2}}(1 + \sqrt{7})\right)^2 \]