Which of the following expression (s) is /are true ?
I. `sqrt(5)+sqrt(5) gt sqrt(2)+sqrt(8)`
II. `sqrt(3)+sqrt(7) gt sqrt(2)+sqrt(8)`
`III. sqrt(5)+sqrt(5) gt sqrt(3)+sqrt(7)`

Options

A. Only I
B. Only I and II
C. Only II
D. All expressions are true

Answer

To determine which of the given expressions are true, we will evaluate each expression one by one.

### Given Expressions:
1. \( \sqrt{5} + \sqrt{5} > \sqrt{2} + \sqrt{8} \)
2. \( \sqrt{3} + \sqrt{7} > \sqrt{2} + \sqrt{8} \)
3. \( \sqrt{5} + \sqrt{5} > \sqrt{3} + \sqrt{7} \)

### Step-by-Step Solution:

#### Expression I: \( \sqrt{5} + \sqrt{5} > \sqrt{2} + \sqrt{8} \)

1. **Calculate \( \sqrt{5} + \sqrt{5} \)**:
\[ \sqrt{5} + \sqrt{5} = 2\sqrt{5} \]

2. **Calculate \( \sqrt{2} + \sqrt{8} \)**:
\[ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \]
Therefore,
\[ \sqrt{2} + \sqrt{8} = \sqrt{2} + 2\sqrt{2} = 3\sqrt{2} \]

3. **Compare \( 2\sqrt{5} \) and \( 3\sqrt{2} \)**:
To compare, we can square both sides (since both are positive):
\[ (2\sqrt{5})^2 = 4 \times 5 = 20 \]
\[ (3\sqrt{2})^2 = 9 \times 2 = 18 \]
Since \( 20 > 18 \), we conclude:
\[ 2\sqrt{5} > 3\sqrt{2} \]
Thus, Expression I is **true**.

#### Expression II: \( \sqrt{3} + \sqrt{7} > \sqrt{2} + \sqrt{8} \)

1. **Calculate \( \sqrt{3} + \sqrt{7} \)**:
We will compare this with \( 3\sqrt{2} \) from the previous calculation.

2. **Square both sides**:
\[ (\sqrt{3} + \sqrt{7})^2 = 3 + 7 + 2\sqrt{3 \cdot 7} = 10 + 2\sqrt{21} \]
\[ (3\sqrt{2})^2 = 9 \times 2 = 18 \]

3. **Compare \( 10 + 2\sqrt{21} \) and \( 18 \)**:
Rearranging gives:
\[ 2\sqrt{21} > 8 \quad \Rightarrow \quad \sqrt{21} > 4 \quad \Rightarrow \quad 21 > 16 \]
Since \( 21 > 16 \), we conclude:
\[ \sqrt{3} + \sqrt{7} > \sqrt{2} + \sqrt{8} \]
Thus, Expression II is **true**.

#### Expression III: \( \sqrt{5} + \sqrt{5} > \sqrt{3} + \sqrt{7} \)

1. **We already know \( \sqrt{5} + \sqrt{5} = 2\sqrt{5} \)**.

2. **Calculate \( \sqrt{3} + \sqrt{7} \)**:
We already calculated this in the previous step:
\[ (\sqrt{3} + \sqrt{7})^2 = 10 + 2\sqrt{21} \]

3. **Compare \( 2\sqrt{5} \) and \( \sqrt{3} + \sqrt{7} \)**:
Square both sides:
\[ (2\sqrt{5})^2 = 20 \]
We need to compare \( 20 \) with \( 10 + 2\sqrt{21} \):
\[ 20 > 10 + 2\sqrt{21} \quad \Rightarrow \quad 10 > 2\sqrt{21} \quad \Rightarrow \quad 5 > \sqrt{21} \]
Since \( \sqrt{21} \approx 4.58 \), we find \( 5 > \sqrt{21} \) is true. Thus:
\[ 2\sqrt{5} > \sqrt{3} + \sqrt{7} \]
Therefore, Expression III is **true**.

### Final Conclusion:
All three expressions are true:
- I: True
- II: True
- III: True