二次根式题50道及答案

Mastering Radicals: Practice Exercises

In mathematics, particularly in algebra, radicals or square roots play a significant role. They are expressions that represent the operation of extracting a root of a number. Let's delve into some practice exercises to enhance your proficiency in dealing with radicals.

Exercise 1: Simplifying Radicals

Simplify the following radicals:

1. \( \sqrt{18} \)

2. \( \sqrt{75} \)

3. \( \sqrt{128} \)

4. \( \sqrt{200} \)

Solution:

1. \( \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \)

2. \( \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \)

3. \( \sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2} = 8\sqrt{2} \)

4. \( \sqrt{200} = \sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2} = 10\sqrt{2} \)

Exercise 2: Operations with Radicals

Perform the indicated operations and simplify the result:

1. \( 3\sqrt{32} 5\sqrt{8} \)

2. \( 2\sqrt{50} 4\sqrt{18} \)

3. \( (\sqrt{27})(\sqrt{12}) \)

4. \( \frac{\sqrt{75}}{\sqrt{3}} \)

Solution:

1. \( 3\sqrt{32} 5\sqrt{8} = 3\sqrt{16 \times 2} 5\sqrt{4 \times 2} = 3 \times 4\sqrt{2} 5 \times 2\sqrt{2} = 12\sqrt{2} 10\sqrt{2} = 22\sqrt{2} \)

2. \( 2\sqrt{50} 4\sqrt{18} = 2\sqrt{25 \times 2} 4\sqrt{9 \times 2} = 2 \times 5\sqrt{2} 4 \times 3\sqrt{2} = 10\sqrt{2} 12\sqrt{2} = 2\sqrt{2} \)

3. \( (\sqrt{27})(\sqrt{12}) = \sqrt{3^3} \times \sqrt{2^2 \times 3} = 3\sqrt{2^2 \times 3} = 3 \times 2\sqrt{3} = 6\sqrt{3} \)

4. \( \frac{\sqrt{75}}{\sqrt{3}} = \sqrt{\frac{75}{3}} = \sqrt{25} = 5 \)

Exercise 3: Rationalizing Denominators

Rationalize the denominators of the following expressions:

1. \( \frac{5}{\sqrt{2}} \)

2. \( \frac{3}{\sqrt{7}} \)

3. \( \frac{2}{\sqrt{5} \sqrt{3}} \)

4. \( \frac{\sqrt{3} 2\sqrt{5}}{\sqrt{3} 2\sqrt{5}} \)

Solution:

1. \( \frac{5}{\sqrt{2}} = \frac{5}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{5\sqrt{2}}{2} \)

2. \( \frac{3}{\sqrt{7}} = \frac{3}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{3\sqrt{7}}{7} \)

3. \( \frac{2}{\sqrt{5} \sqrt{3}} = \frac{2(\sqrt{5} \sqrt{3})}{(\sqrt{5} \sqrt{3})(\sqrt{5} \sqrt{3})} = \frac{2(\sqrt{5} \sqrt{3})}{5 3} = \frac{2(\sqrt{5} \sqrt{3})}{2} = \sqrt{5} \sqrt{3} \)

4. \( \frac{\sqrt{3} 2\sqrt{5}}{\sqrt{3} 2\sqrt{5}} = \frac{(\sqrt{3} 2\sqrt{5})(\sqrt{3} 2\sqrt{5})}{(\sqrt{3} 2\sqrt{5})(\sqrt{3} 2\sqrt{5})} = \frac{3 2\sqrt{15} 2\sqrt{15} 20}{3 2\sqrt{15} 2\sqrt{15} 20} = \frac{23 4\sqrt{15}}{17} = \frac{23}{17} \frac{4\sqrt{15}}{17} \)

These exercises cover a range of operations involving radicals, from simplification to addition, subtraction, multiplication, division, and rationalization. Practicing these concepts will strengthen your understanding and proficiency in working with radicals. Keep practicing to master these skills!