Câu 70:

Tìm giá trị các biểu thức sau bằng cách biến đổi, rút gọn thích hợp:

a) \(\sqrt{\frac{25}{81}, \frac{16}{49}, \frac{196}{9}}\)

b) \(\sqrt{3 \frac{1}{16} \cdot 2 \frac{14}{25} \cdot 2 \frac{34}{81}}\)

c) \(\frac{\sqrt{640} \cdot \sqrt{34,3}}{\sqrt{567}}\)

d) \(\sqrt{21,6} \cdot \sqrt{810} \cdot \sqrt{11^{2}-5^{2}}\)

Lời giải:

a) \(\sqrt{\frac{25}{81} \cdot \frac{16}{49} \cdot \frac{196}{9}}=\sqrt{\frac{25}{81}} \cdot \sqrt{\frac{16}{49}} \cdot \sqrt{\frac{196}{9}}\)

\(= \frac{\sqrt{25}}{\sqrt{81}} \cdot \frac{\sqrt{16}}{\sqrt{49}} \cdot \frac{\sqrt{196}}{\sqrt{9}}=\frac{\sqrt{5^{2}}}{\sqrt{9^{2}}} \cdot \frac{\sqrt{4^{2}}}{\sqrt{7^{2}}} \cdot \frac{\sqrt{14^{2}}}{\sqrt{3^{2}}}\)

\(= \frac{5.4 .14}{9.7 .3}=\frac{5.4 .2}{9.3}=\frac{40}{27}\)

b) \(\sqrt{3 \frac{1}{16} \cdot 2 \frac{14}{25} \cdot 2 \frac{34}{81}}=\sqrt{\frac{49}{16} \cdot \frac{64}{25} \cdot \frac{196}{81}}\)

\(= \sqrt{\frac{49}{16}} \cdot \sqrt{\frac{64}{25}} \cdot \sqrt{\frac{196}{81}}\)

\(= \frac{\sqrt{7^{2}}}{\sqrt{4^{2}}} \cdot \frac{\sqrt{8^{2}}}{\sqrt{5^{2}}} \cdot \frac{\sqrt{14^{2}}}{\sqrt{9^{2}}}\)

\(= \frac{7.8 .14}{4.5 .9}=\frac{7.4 .7}{5.9}=\frac{196}{45}\)

c) \(\frac{\sqrt{640} \cdot \sqrt{34,3}}{\sqrt{567}}=\sqrt{\frac{640.34,3}{567}}=\sqrt{\frac{64.343}{567}}\)

\(=\sqrt{\frac{64.49 .7}{81.7}}=\sqrt{\frac{64.49}{81}}=\frac{8.7}{9}=\frac{56}{9}\)

d) \(\sqrt{21,6} \cdot \sqrt{810} \cdot \sqrt{11^{2}-5^{2}}\)

\(=\sqrt{21,6 \cdot 810} \cdot \sqrt{(11-5)(11+5)}\)

\(=\sqrt{216 \cdot 81} \cdot \sqrt{6 \cdot 16}=\sqrt{216 \cdot 6 \cdot 81 \cdot 16}\)

\(=\sqrt{1296.81 \cdot 16}=\sqrt{36^{2} \cdot 9^{2} \cdot 4^{2}}\)

\(=\sqrt{(36.9 \cdot 4)^{2}}=1296\)