Which is the correct statement :-

Options

A. `sqrt2+sqrt5=sqrt7`
B. `sqrt2+sqrt5lesqrt7`
C. `sqrt2+sqrt5ltsqrt7`
D. `sqrt2+sqrt5gtsqrt7`

Answer

To solve the question, we need to analyze the given statements regarding the sum of square roots and compare them with the square root of 7. Let's go through the solution step by step.

### Step 1: Analyze the Statements
We have four options to evaluate:
1. \( \sqrt{2} + \sqrt{5} = \sqrt{7} \)
2. \( \sqrt{2} + \sqrt{5} \leq \sqrt{7} \)
3. \( \sqrt{2} + \sqrt{5} < \sqrt{7} \)
4. \( \sqrt{2} + \sqrt{7} > \sqrt{7} \)

### Step 2: Square Both Sides
To compare \( \sqrt{2} + \sqrt{5} \) with \( \sqrt{7} \), we will square both sides of the first statement:
\[ (\sqrt{2} + \sqrt{5})^2 \quad \text{and} \quad (\sqrt{7})^2 \]

### Step 3: Expand the Left Side
Using the formula \( (a + b)^2 = a^2 + b^2 + 2ab \):
\[ (\sqrt{2} + \sqrt{5})^2 = (\sqrt{2})^2 + (\sqrt{5})^2 + 2 \cdot \sqrt{2} \cdot \sqrt{5} \]
Calculating this gives:
\[ = 2 + 5 + 2\sqrt{10} = 7 + 2\sqrt{10} \]

### Step 4: Calculate the Right Side
The right side is simply:
\[ (\sqrt{7})^2 = 7 \]

### Step 5: Compare the Two Sides
Now we compare:
\[ 7 + 2\sqrt{10} \quad \text{and} \quad 7 \]
Since \( 2\sqrt{10} \) is a positive number, we can conclude:
\[ 7 + 2\sqrt{10} > 7 \]
Thus, we have:
\[ \sqrt{2} + \sqrt{5} > \sqrt{7} \]

### Step 6: Evaluate the Options
From our analysis, we see that:
- The first statement is false.
- The second statement is false.
- The third statement is false.
- The fourth statement \( \sqrt{2} + \sqrt{7} > \sqrt{7} \) is true since \( \sqrt{2} > 0 \).

Therefore, the correct statement is:
**Option 4: \( \sqrt{2} + \sqrt{7} > \sqrt{7} \)**.