Câu 71:

Rút gọn các biểu thức sau:

a) \((\sqrt{8}-3 \sqrt{2}+\sqrt{10}) \sqrt{2}-\sqrt{5}\)

b) \(0,2 \sqrt{(-10)^{2} \cdot 3}+2 \sqrt{(\sqrt{3}-\sqrt{5})^{2}}\)

c) \(\left(\frac{1}{2} \sqrt{\frac{1}{2}}-\frac{3}{2} \cdot \sqrt{2}+\frac{4}{5} \cdot \sqrt{200}\right): \frac{1}{8}\)

d) \(2 \sqrt{(\sqrt{2}-3)^{2}}+\sqrt{2 \cdot(-3)^{2}}-5 \sqrt{(-1)^{4}}\)

Lời giải:

a) \((\sqrt{8}-3 \sqrt{2}+\sqrt{10}) \sqrt{2}-\sqrt{5}\)

\(=\left(\sqrt{2^{2} \cdot 2}-3 \sqrt{2}+\sqrt{5 \cdot 2}\right) \sqrt{2}-\sqrt{5}\)

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

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

\(=(\sqrt{5}-1) \cdot 2-\sqrt{5}\)

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

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

b) \(0,2 \sqrt{(-10)^{2} \cdot 3}+2 \sqrt{(\sqrt{3}-\sqrt{5})^{2}}\)

\(=0,2 \cdot 10 \cdot \sqrt{3}+2 \cdot|\sqrt{3}-\sqrt{5}|\)

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

c) \(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{2}}-\frac{3}{2} \cdot \sqrt{2}+\frac{4}{5} \sqrt{200}\right): \frac{1}{8}\)

\(=\left(\frac{1}{2} \sqrt{\frac{1}{2}}-3 \sqrt{\frac{2}{2^{2}}}+\frac{4}{5} \sqrt{\frac{400}{2}}\right): \frac{1}{8}\)

\(=\left(\frac{1}{2} \sqrt{\frac{1}{2}}-3 \sqrt{\frac{1}{2}}+\frac{4}{5} \cdot 20 \sqrt{\frac{1}{2}}\right): \frac{1}{8}\)

\(=\left(\frac{1}{2} \sqrt{\frac{1}{2}}-3 \sqrt{\frac{1}{2}}+16 \sqrt{\frac{1}{2}}\right) \cdot 8\)

\(=\frac{27}{2} \sqrt{\frac{1}{2}} \cdot 8=27.2 \sqrt{2}=54 \sqrt{2}\)

d) \(2 \sqrt{(\sqrt{2}-3)^{2}}+\sqrt{2 \cdot(-3)^{2}}-5 \sqrt{(-1)^{4}}\)

\(=2|\sqrt{2}-3|+3 \sqrt{2}-5 \sqrt{1}\)

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

\(=1+\sqrt{2}\)