Question

Arrange the following expressions in ascending order: \[ \sqrt{401} - \sqrt{399}, \quad \sqrt{301} - \sqrt{299}, \quad \sqrt{201} - \sqrt{199}. \]

• A. \(\sqrt{401} - \sqrt{399}<\sqrt{301} - \sqrt{299}<\sqrt{201} - \sqrt{199}\)
• B. \(\sqrt{201} - \sqrt{199}<\sqrt{301} - \sqrt{299}<\sqrt{401} - \sqrt{399}\)
• C. \(\sqrt{301} - \sqrt{299}<\sqrt{201} - \sqrt{199}<\sqrt{401} - \sqrt{399}\)
• D. \(\sqrt{201} - \sqrt{199}<\sqrt{401} - \sqrt{399}<\sqrt{301} - \sqrt{299}\)

Hint:

For such problems, approximate the square roots to compare the expressions or use the formula: \[ \sqrt{a} - \sqrt{b} \approx \frac{a - b}{\sqrt{a} + \sqrt{b}}. \]

Answer 1

The Correct Option is B

We calculate the approximate values of each expression: \(\sqrt{401} - \sqrt{399} \approx 0.050\) \(\sqrt{301} - \sqrt{299} \approx 0.058\) \(\sqrt{201} - \sqrt{199} \approx 0.070\) From these calculations, the order of the expressions in ascending order is: \[ \sqrt{201} - \sqrt{199}<\sqrt{301} - \sqrt{299}<\sqrt{401} - \sqrt{399}. \]