Which of the two expressions , `sqrt(11)-sqrt(10)` and `sqrt(12)-sqrt(11)` is greater?

Answer

To determine which of the two expressions, \( \sqrt{11} - \sqrt{10} \) and \( \sqrt{12} - \sqrt{11} \), is greater, we can rationalize both expressions and compare their values.

### Step-by-Step Solution:

**Step 1: Rationalize the first expression \( \sqrt{11} - \sqrt{10} \)**

To rationalize \( \sqrt{11} - \sqrt{10} \), we multiply the numerator and denominator by \( \sqrt{11} + \sqrt{10} \):

\[ \sqrt{11} - \sqrt{10} = \frac{(\sqrt{11} - \sqrt{10})(\sqrt{11} + \sqrt{10})}{\sqrt{11} + \sqrt{10}} = \frac{11 - 10}{\sqrt{11} + \sqrt{10}} = \frac{1}{\sqrt{11} + \sqrt{10}} \]

**Step 2: Rationalize the second expression \( \sqrt{12} - \sqrt{11} \)**

Similarly, we rationalize \( \sqrt{12} - \sqrt{11} \) by multiplying the numerator and denominator by \( \sqrt{12} + \sqrt{11} \):

\[ \sqrt{12} - \sqrt{11} = \frac{(\sqrt{12} - \sqrt{11})(\sqrt{12} + \sqrt{11})}{\sqrt{12} + \sqrt{11}} = \frac{12 - 11}{\sqrt{12} + \sqrt{11}} = \frac{1}{\sqrt{12} + \sqrt{11}} \]

**Step 3: Compare the two rationalized expressions**

Now we have:

1. \( \sqrt{11} - \sqrt{10} = \frac{1}{\sqrt{11} + \sqrt{10}} \)
2. \( \sqrt{12} - \sqrt{11} = \frac{1}{\sqrt{12} + \sqrt{11}} \)

To compare these two fractions, we note that:

- \( \sqrt{11} + \sqrt{10} \) is less than \( \sqrt{12} + \sqrt{11} \) because \( \sqrt{12} > \sqrt{11} \).
- Therefore, since the numerator (1) is the same, the fraction with the smaller denominator will be larger.

Thus, we conclude:

\[ \sqrt{11} - \sqrt{10} > \sqrt{12} - \sqrt{11} \]

### Final Answer:
\(\sqrt{11} - \sqrt{10}\) is greater than \(\sqrt{12} - \sqrt{11}\).

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