1.6 Rút gọn các biểu thức sau:

a) \(5 \sqrt{1 / 5} + \frac{1}{2} \sqrt{20} + \sqrt{5}\)

Ta có:

• \(\sqrt{1 / 5} = \frac{1}{\sqrt{5}} \Rightarrow 5 \sqrt{1 / 5} = \frac{5}{\sqrt{5}} = \sqrt{5} \cdot \frac{5}{\sqrt{5}} = 5\)
• \(\frac{1}{2} \sqrt{20} = \frac{1}{2} \cdot \sqrt{4 \cdot 5} = \frac{1}{2} \cdot 2 \sqrt{5} = \sqrt{5}\)
• \(\sqrt{5}\) giữ nguyên

👉 Vậy:
\(5 \sqrt{1 / 5} + \frac{1}{2} \sqrt{20} + \sqrt{5} = 5 + \sqrt{5} + \sqrt{5} = 5 + 2 \sqrt{5}\)

b) \(1 / \sqrt{2} + \sqrt{4.5} + \sqrt{12.5}\)

Ta có:

• \(\sqrt{4.5} = \sqrt{\frac{9}{2}} = \frac{3}{\sqrt{2}}\)
• \(\sqrt{12.5} = \sqrt{\frac{25}{2}} = \frac{5}{\sqrt{2}}\)
• \(\frac{1}{\sqrt{2}}\) giữ nguyên

👉 Cộng lại:

\(\frac{1}{\sqrt{2}} + \frac{3}{\sqrt{2}} + \frac{5}{\sqrt{2}} = \frac{9}{\sqrt{2}} \Rightarrow \frac{9 \sqrt{2}}{2}\)

c) \(\sqrt{20} - \sqrt{45} + 3 \sqrt{18} + \sqrt{72}\)

Ta rút gọn từng căn:

• \(\sqrt{20} = \sqrt{4 \cdot 5} = 2 \sqrt{5}\)
• \(\sqrt{45} = \sqrt{9 \cdot 5} = 3 \sqrt{5}\)
• \(\sqrt{18} = \sqrt{9 \cdot 2} = 3 \sqrt{2} \Rightarrow 3 \sqrt{18} = 9 \sqrt{2}\)
• \(\sqrt{72} = \sqrt{36 \cdot 2} = 6 \sqrt{2}\)

👉 Vậy:

\(2 \sqrt{5} - 3 \sqrt{5} + 9 \sqrt{2} + 6 \sqrt{2} = - \sqrt{5} + 15 \sqrt{2}\)

d) \(\sqrt{20} - \sqrt{45} + 3 \sqrt{18} + \sqrt{72}\)

(giống y câu c, kết quả vẫn là: \(- \sqrt{5} + 15 \sqrt{2}\))

e) \(\left(\right. \sqrt{6} + \sqrt{5} \left.\right)^{2} - \sqrt{120}\)

Ta khai triển:

• \(\left(\right. \sqrt{6} + \sqrt{5} \left.\right)^{2} = 6 + 2 \sqrt{30} + 5 = 11 + 2 \sqrt{30}\)
• \(\sqrt{120} = \sqrt{4 \cdot 30} = 2 \sqrt{30}\)

👉 Vậy:

\(11 + 2 \sqrt{30} - 2 \sqrt{30} = 11\)