23.概念学习:规定,求若干个相同的有理数(均不等于0)的除法运算叫作除方.类比有理数的乘方,我们把$2÷2÷2$记作$2^{③}$,读作“2的圈3次方”,$(-3)÷(-3)÷(-3)÷(-3)$记作$(-3)^{④}$,读作“-3的圈4次方”.一般地,把$n$个$a(a≠0)$相除记作$a^{ⓝ}$,读作“$a$的圈$n$次方”.
(1)初步探究:直接写出计算结果:$2^{③}=$
(2)深入思考:我们会发现,有理数的除方运算可以转化为乘方运算.
例如:$(-3)^{④}=(-3)÷(-3)÷(-3)÷(-3)=(-3)×(-\dfrac{1}{3})×(-\dfrac{1}{3})×(-\dfrac{1}{3})=(-\dfrac{1}{3})^{2}=(\dfrac{1}{3})^{2}$.
类比上面的计算,将下列运算结果直接写成幂的形式:$7^{④}=$
(3)利用上述规律计算:$2^{2}÷(-\dfrac{1}{3})^{③}×(-2)^{④}-(-\dfrac{1}{3})^{⑥}÷3^{3}$.
(1)初步探究:直接写出计算结果:$2^{③}=$
$\dfrac{1}{2}$
;$(-3)^{④}=$$\dfrac{1}{9}$
.(2)深入思考:我们会发现,有理数的除方运算可以转化为乘方运算.
例如:$(-3)^{④}=(-3)÷(-3)÷(-3)÷(-3)=(-3)×(-\dfrac{1}{3})×(-\dfrac{1}{3})×(-\dfrac{1}{3})=(-\dfrac{1}{3})^{2}=(\dfrac{1}{3})^{2}$.
类比上面的计算,将下列运算结果直接写成幂的形式:$7^{④}=$
$(\dfrac{1}{7})^2$
;$(-\dfrac{1}{5})^{⑥}=$$5^4$
.(3)利用上述规律计算:$2^{2}÷(-\dfrac{1}{3})^{③}×(-2)^{④}-(-\dfrac{1}{3})^{⑥}÷3^{3}$.
答案
(1)$\dfrac{1}{2}\quad \dfrac{1}{9}$
(2)$7^{④}=7÷7÷7÷7=7×\dfrac{1}{7}×\dfrac{1}{7}×\dfrac{1}{7}=(\dfrac{1}{7})^2$;
$(-\dfrac{1}{5})^{⑥}=(-\dfrac{1}{5})÷(-\dfrac{1}{5})÷(-\dfrac{1}{5})÷(-\dfrac{1}{5})÷(-\dfrac{1}{5})÷(-\dfrac{1}{5})$
$=(-\dfrac{1}{5})×(-5)×(-5)×(-5)×(-5)×(-5)$
$=(-5)^4=5^4$.
(3)$2^2÷(-\dfrac{1}{3})^{③}×(-2)^{④}-(-\dfrac{1}{3})^{⑥}÷3^3$
$=4÷(-3)×(-\dfrac{1}{2})^2-(-3)^4÷3^3$
$=4×(-\dfrac{1}{3})×\dfrac{1}{4}-3^4÷3^3$
$=-\dfrac{1}{3}-3$
$=-\dfrac{10}{3}$.
(2)$7^{④}=7÷7÷7÷7=7×\dfrac{1}{7}×\dfrac{1}{7}×\dfrac{1}{7}=(\dfrac{1}{7})^2$;
$(-\dfrac{1}{5})^{⑥}=(-\dfrac{1}{5})÷(-\dfrac{1}{5})÷(-\dfrac{1}{5})÷(-\dfrac{1}{5})÷(-\dfrac{1}{5})÷(-\dfrac{1}{5})$
$=(-\dfrac{1}{5})×(-5)×(-5)×(-5)×(-5)×(-5)$
$=(-5)^4=5^4$.
(3)$2^2÷(-\dfrac{1}{3})^{③}×(-2)^{④}-(-\dfrac{1}{3})^{⑥}÷3^3$
$=4÷(-3)×(-\dfrac{1}{2})^2-(-3)^4÷3^3$
$=4×(-\dfrac{1}{3})×\dfrac{1}{4}-3^4÷3^3$
$=-\dfrac{1}{3}-3$
$=-\dfrac{10}{3}$.
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