Which is greatest `(sqrt(7)+sqrt(10))` ro` sqrt(3)+sqrt(19))`?

Options

A. `sqrt(7)+sqrt(10)`
B. `sqrt(3)+sqrt(19)`
C. both are equal
D. None of these

Answer

To determine which expression is greater between \((\sqrt{7} + \sqrt{10})\) and \((\sqrt{3} + \sqrt{19})\), we can follow these steps:

### Step 1: Define the expressions
Let:
- \( x = \sqrt{7} + \sqrt{10} \)
- \( y = \sqrt{3} + \sqrt{19} \)

### Step 2: Square both expressions
To compare \(x\) and \(y\), we will square both expressions. This is because if \(x^2 > y^2\), then \(x > y\) (and vice versa).

Calculating \(x^2\):
\[ x^2 = (\sqrt{7} + \sqrt{10})^2 = (\sqrt{7})^2 + 2(\sqrt{7})(\sqrt{10}) + (\sqrt{10})^2 \]
\[ x^2 = 7 + 10 + 2\sqrt{70} = 17 + 2\sqrt{70} \]

Calculating \(y^2\):
\[ y^2 = (\sqrt{3} + \sqrt{19})^2 = (\sqrt{3})^2 + 2(\sqrt{3})(\sqrt{19}) + (\sqrt{19})^2 \]
\[ y^2 = 3 + 19 + 2\sqrt{57} = 22 + 2\sqrt{57} \]

### Step 3: Compare \(x^2\) and \(y^2\)
Now we need to compare \(x^2\) and \(y^2\):
- \(x^2 = 17 + 2\sqrt{70}\)
- \(y^2 = 22 + 2\sqrt{57}\)

### Step 4: Analyze the comparison
We can compare the constant parts first:
- The constant part of \(x^2\) is \(17\)
- The constant part of \(y^2\) is \(22\)

Since \(22 > 17\), it suggests that \(y^2\) might be greater than \(x^2\). However, we must also consider the square root terms.

### Step 5: Approximate the square root terms
To make a more precise comparison, we can approximate the square roots:
- \(\sqrt{70} \approx 8.37\) (since \(8.37^2 \approx 70\))
- \(\sqrt{57} \approx 7.55\) (since \(7.55^2 \approx 57\))

Now substituting these approximations:
- \(2\sqrt{70} \approx 2 \times 8.37 \approx 16.74\)
- \(2\sqrt{57} \approx 2 \times 7.55 \approx 15.1\)

### Step 6: Calculate approximate values of \(x^2\) and \(y^2\)
Now we can calculate:
- \(x^2 \approx 17 + 16.74 = 33.74\)
- \(y^2 \approx 22 + 15.1 = 37.1\)

### Step 7: Conclusion
Since \(y^2 \approx 37.1\) is greater than \(x^2 \approx 33.74\), we conclude that:
\[ y > x \]
Thus, \(\sqrt{3} + \sqrt{19}\) is greater than \(\sqrt{7} + \sqrt{10}\).

### Final Answer
\(\sqrt{3} + \sqrt{19}\) is the greater expression.

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