Giải Bài 55 trang 26 sách bài tập toán 7 tập 1 - Cánh diều

Đề bài

Sắp xếp các số sau theo thứ tự tăng dần:

a) \({\left( {\dfrac{{22}}{{21}}} \right)^{18}};{\rm{ }}{\left( {\dfrac{{22}}{{21}}} \right)^{21}};{\rm{ }}{\left( {\dfrac{{22}}{{21}}} \right)^{20}};{\rm{ }}{\left( {\dfrac{{22}}{{21}}} \right)^{22}};{\rm{ }}\dfrac{{22}}{{21}}\);

b) \({(0,1)^{21}};{\rm{ (}} - {\rm{0,1}}{{\rm{)}}^{20}};{\rm{ (0,1}}{{\rm{)}}^{22}};{\rm{ (}} - 0,1{)^{19}};{\rm{ 0}}\).

Lời giải chi tiết

a) \({\left( {\dfrac{{22}}{{21}}} \right)^{18}};{\rm{ }}{\left( {\dfrac{{22}}{{21}}} \right)^{21}};{\rm{ }}{\left( {\dfrac{{22}}{{21}}} \right)^{20}};{\rm{ }}{\left( {\dfrac{{22}}{{21}}} \right)^{22}};{\rm{ }}\dfrac{{22}}{{21}}\);

Ta có: \(\dfrac{{22}}{{21}} > 1\) nên \(\dfrac{{22}}{{21}} < {\left( {\dfrac{{22}}{{21}}} \right)^{18}} < {\left( {\dfrac{{22}}{{21}}} \right)^{20}} < {\left( {\dfrac{{22}}{{21}}} \right)^{21}} < {\left( {\dfrac{{22}}{{21}}} \right)^{22}}\).

Các số theo thứ tự tăng dần là: \(\dfrac{{22}}{{21}};{\rm{ }}{\left( {\dfrac{{22}}{{21}}} \right)^{18}};{\rm{ }}{\left( {\dfrac{{22}}{{21}}} \right)^{20}};{\rm{ }}{\left( {\dfrac{{22}}{{21}}} \right)^{21}};{\rm{ }}{\left( {\dfrac{{22}}{{21}}} \right)^{22}}\).

b) \({(0,1)^{21}};{\rm{ (}} - {\rm{0,1}}{{\rm{)}}^{20}};{\rm{ (0,1}}{{\rm{)}}^{22}};{\rm{ (}} - 0,1{)^{19}};{\rm{ 0}}\).

Cách 1: Ta có: \( - 0,1 < 0 < 0,1\) nên: \({( - 0,1)^{19}} < 0\).

Ta xét: \({(0,1)^{21}};{\rm{ (}} - {\rm{0,1}}{{\rm{)}}^{20}};{\rm{ (0,1}}{{\rm{)}}^{22}}\)có: \(\begin{array}{l}{(0,1)^{21}} = {\left( {\dfrac{1}{{10}}} \right)^{21}} = \dfrac{1}{{{{10}^{21}}}}\\{( - 0,1)^{20}} = {(0,1)^{20}} = {\left( {\dfrac{1}{{10}}} \right)^{20}} = \dfrac{1}{{{{10}^{20}}}}\\{(0,1)^{22}} = {\left( {\dfrac{1}{{10}}} \right)^{22}} = \dfrac{1}{{{{10}^{22}}}}\end{array}\)

Mà \({10^{20}} < {10^{21}} < {10^{22}} \Rightarrow \dfrac{1}{{{{10}^{20}}}} > \dfrac{1}{{{{10}^{21}}}} > \dfrac{1}{{{{10}^{22}}}}\) nên: \({{\rm{(}} - {\rm{0,1)}}^{20}}{\rm{ >  }}{(0,1)^{21}} > {{\rm{(0,1)}}^{22}}\).

Vậy sắp xếp các số theo thứ tự tăng dần là: \({{\rm{(}} - 0,1)^{19}};{\rm{ 0; (0,1}}{{\rm{)}}^{22}};{\rm{ }}{(0,1)^{21}};{\rm{ (}} - {\rm{0,1}}{{\rm{)}}^{20}}{\rm{ }}\). 

Cách 2: Ta có: \( - 0,1 < 0 < 0,1\) nên: \({( - 0,1)^{19}} < 0\).

\((-0,1)^{20}=(0,1)^{20}\)

Vì \(0< 0,1 < 1\) nên \((0,1)^{20}> (0,1)^{21}>(0,1)^{22}>0\) hay \((-0,1)^{20}> (0,1)^{21}>(0,1)^{22}\)

Vậy sắp xếp các số theo thứ tự tăng dần là: \({{\rm{(}} - 0,1)^{19}};{\rm{ 0; (0,1}}{{\rm{)}}^{22}};{\rm{ }}{(0,1)^{21}};{\rm{ (}} - {\rm{0,1}}{{\rm{)}}^{20}}{\rm{ }}\).