Which one among `sqrt10 + sqrt4, sqrt11 + sqrt3 , sqrt7 + sqrt7` is the smallest number ?

Options

A. `sqrt10 + sqrt4`
B. `sqrt11 + sqrt3`
C. `sqrt7 + sqrt7`
D. All are equal

Answer

To determine which among the three expressions \( \sqrt{10} + \sqrt{4} \), \( \sqrt{11} + \sqrt{3} \), and \( \sqrt{7} + \sqrt{7} \) is the smallest, we can follow these steps:

### Step 1: Calculate each expression
1. **First Expression: \( \sqrt{10} + \sqrt{4} \)**
- \( \sqrt{4} = 2 \)
- Therefore, \( \sqrt{10} + \sqrt{4} = \sqrt{10} + 2 \)

2. **Second Expression: \( \sqrt{11} + \sqrt{3} \)**
- This remains as \( \sqrt{11} + \sqrt{3} \)

3. **Third Expression: \( \sqrt{7} + \sqrt{7} \)**
- This simplifies to \( 2\sqrt{7} \)

### Step 2: Square each expression to compare their sizes
1. **Square of the First Expression:**
\[ (\sqrt{10} + 2)^2 = (\sqrt{10})^2 + 2 \cdot \sqrt{10} \cdot 2 + 2^2 = 10 + 4\sqrt{10} + 4 = 14 + 4\sqrt{10} \]

2. **Square of the Second Expression:**
\[ (\sqrt{11} + \sqrt{3})^2 = (\sqrt{11})^2 + 2 \cdot \sqrt{11} \cdot \sqrt{3} + (\sqrt{3})^2 = 11 + 2\sqrt{33} + 3 = 14 + 2\sqrt{33} \]

3. **Square of the Third Expression:**
\[ (2\sqrt{7})^2 = 4 \cdot 7 = 28 \]

### Step 3: Compare the squared values
- **First Expression:** \( 14 + 4\sqrt{10} \)
- **Second Expression:** \( 14 + 2\sqrt{33} \)
- **Third Expression:** \( 28 \)

### Step 4: Estimate the square roots
1. **Estimate \( \sqrt{10} \):** Approximately \( 3.16 \)
- Thus, \( 4\sqrt{10} \approx 4 \times 3.16 = 12.64 \)
- So, \( 14 + 4\sqrt{10} \approx 14 + 12.64 = 26.64 \)

2. **Estimate \( \sqrt{33} \):** Approximately \( 5.74 \)
- Thus, \( 2\sqrt{33} \approx 2 \times 5.74 = 11.48 \)
- So, \( 14 + 2\sqrt{33} \approx 14 + 11.48 = 25.48 \)

3. **Third Expression:** \( 28 \)

### Step 5: Conclusion
Now we compare the approximated values:
- First Expression: \( \approx 26.64 \)
- Second Expression: \( \approx 25.48 \)
- Third Expression: \( 28 \)

The smallest value is from the **Second Expression** \( \sqrt{11} + \sqrt{3} \).

### Final Answer
The smallest number among \( \sqrt{10} + \sqrt{4} \), \( \sqrt{11} + \sqrt{3} \), and \( \sqrt{7} + \sqrt{7} \) is \( \sqrt{11} + \sqrt{3} \).

---