Tính một cách hợp lí:
a) \(\dfrac{{ - 5}}{7}.\dfrac{2}{{11}} + \dfrac{{ - 5}}{7}.\dfrac{9}{{11}} + \dfrac{5}{7}\);
b) \(\left[ {\left( {\dfrac{{ - 3}}{8} + \dfrac{{11}}{{23}}} \right):\dfrac{5}{9} + \left( {\dfrac{{ - 5}}{8} + \dfrac{{12}}{{23}}} \right):\dfrac{5}{9}} \right].\dfrac{{ - 11}}{{325}}\);
c*) \(\dfrac{{{{15}^5}}}{{{5^5}}} - {( - 0,25)^2}{.4^2}\);
d*) \( - \dfrac{{{2^{15}}{{.9}^4}}}{{{6^6}{{.8}^3}}} + 0,75.\dfrac{{ - 1}}{2} + 0,375\).
Ta thực hiện phép tính bằng cách tính nhanh.
Lưu ý:
\(\dfrac{{{x^m}}}{{{y^m}}} = {\left( {\dfrac{x}{y}} \right)^m}{\rm{ }}(y \ne 0)\).
\({(x.y)^m} = {x^m}.{y^m}\).
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\({x^m}.{x^n} = {x^{m + n}};{\rm{ }}\dfrac{{{x^m}}}{{{x^n}}} = {x^{m - n}}{\rm{ (}}m > n)\).
a)
\(\dfrac{{ - 5}}{7}.\dfrac{2}{{11}} + \dfrac{{ - 5}}{7}.\dfrac{9}{{11}} + \dfrac{5}{7} \\= \dfrac{{ - 5}}{7}.\left( {\dfrac{2}{{11}} + \dfrac{9}{{11}}} -1\right) \\= \dfrac{{ - 5}}{7}.(\dfrac{11}{11}-1) = \dfrac{{ - 5}}{7} .0 = 0\)
b)
\(\begin{array}{l}\left[ {\left( {\dfrac{{ - 3}}{8} + \dfrac{{11}}{{23}}} \right):\dfrac{5}{9} + \left( {\dfrac{{ - 5}}{8} + \dfrac{{12}}{{23}}} \right):\dfrac{5}{9}} \right].\dfrac{{ - 11}}{{325}}\\ = \left[ {\left( {\dfrac{{ - 3}}{8} + \dfrac{{11}}{{23}} + \dfrac{{ - 5}}{8} + \dfrac{{12}}{{23}}} \right):\dfrac{5}{9}} \right].\dfrac{{ - 11}}{{325}}\\{\rm{ }} \\= \left[ {\left( {\dfrac{{ - 3}}{8} +\dfrac{{ - 5}}{8}+ \dfrac{{11}}{{23}} + \dfrac{{12}}{{23}}} \right):\dfrac{5}{9}} \right].\dfrac{{ - 11}}{{325}}\\{\rm{ }} \\= \left[ {( - 1 + 1):\dfrac{5}{9}} \right].\dfrac{{ - 11}}{{325}} = \left( {0:\dfrac{5}{9}} \right).\dfrac{{ - 11}}{{325}} \\= 0.\dfrac{{ - 11}}{{325}} = 0\end{array}\)
c*)
\(\dfrac{{{{15}^5}}}{{{5^5}}} - {( - 0,25)^2}{.4^2} = {\left( {\dfrac{{15}}{5}} \right)^5} - {\left( {\dfrac{{ - 1}}{4}} \right)^2}{.4^2} \\= {3^5} - \dfrac{1}{{{4^2}}}{.4^2} \\= {3^5} - 1 = 243 - 1 = 242\)
d*)
\(\begin{array}{l} - \dfrac{{{2^{15}}{{.9}^4}}}{{{6^6}{{.8}^3}}} + 0,75.\dfrac{{ - 1}}{2} + 0,375\\ = - \dfrac{{{2^{15}}{{.3}^{{2^4}}}}}{{{{(2.3)}^6}{{.2}^{{3^3}}}}} + ( - 0,375) + 0,375\\{\rm{ }} \\= - \dfrac{{{2^{15}}{{.3}^8}}}{{{2^6}{{.3}^6}{{.2}^9}}}+ \left[ {( - 0,375) + 0,375} \right]\\ = - \dfrac{{{2^{15}}{{.3}^8}}}{{{2^{15}}{{.3}^6}}} \\ = - \dfrac{{{3^8}}}{{{3^6}}} = - {3^2} = - 9\end{array}\).