Câu 58:

Rút gọn các biểu thức sau:

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

b) \(\sqrt{\frac{1}{2}}+\sqrt{4,5}+\sqrt{12,5}\)

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

d) \(0,1 \cdot \sqrt{200}+2 \cdot \sqrt{0,08}+0,4 \cdot \sqrt{50}\)

Lời giải:

a) \(5 \sqrt{\frac{1}{5}}+\frac{1}{2} \sqrt{20}+\sqrt{5}=\sqrt{5^{2} \cdot \frac{1}{5}}+\sqrt{\frac{1}{2^{2}} \cdot 20}+\sqrt{5}\)

\(=\sqrt{5}+\sqrt{5}+\sqrt{5}=3 \sqrt{5}\)

b) \(\sqrt{\frac{1}{2}}+\sqrt{4,5}+\sqrt{12,5}=\sqrt{\frac{1}{2}}+\sqrt{\frac{9}{2}}+\sqrt{\frac{25}{2}}\)

\(=\sqrt{\frac{1}{2}}+\sqrt{3^{2} \cdot \frac{1}{2}}+\sqrt{5^{2} \cdot \frac{1}{2}}=\sqrt{\frac{1}{2}}+3 \sqrt{\frac{1}{2}}+5 \sqrt{\frac{1}{2}}\)

\(=9 \sqrt{\frac{1}{2}}=9 \frac{\sqrt{1}}{\sqrt{2}}=9 \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}=\frac{9}{2} \sqrt{2}\)

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

\(= \sqrt{4} .5-\sqrt{9} .5+3 \sqrt{9} .2+\sqrt{3} 6.2\)

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

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

d) \(0,1 \cdot \sqrt{200}+2 \cdot \sqrt{0,08}+0,4 \cdot \sqrt{50}\)

\(= 0,1 \cdot \sqrt{100 \cdot 2}+2 \sqrt{\frac{8}{100}}+0,4 \cdot \sqrt{25 \cdot 2}\)

\(= 0,1 \cdot 10 \sqrt{2}+2 \frac{\sqrt{4 \cdot 2}}{\sqrt{100}}+0,4 \cdot 5 \sqrt{2}\)

\(= \sqrt{2}+\frac{2 \cdot 2 \cdot \sqrt{2}}{10}+2 \sqrt{2}\)

\(= \frac{17 \sqrt{2}}{5}\)