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指数幂的运算

开发技术 开发技术 5小时前 2次浏览

前言

运算法则

  • 实数指数幂的运算性质如下:此时(a>0)(b>0)(m,nin R)

公式:(a^mcdot a^n=a^{m+n})((a^m)^n=(a^n)^m=a^{mn})((acdot b)^n=a^ncdot b^n)((cfrac{a}{b})^n=cfrac{a^n}{b^n}=a^ncdot b^{-n})

  • 指数幂运算的一般原则

(1).有括号的先算括号里的,无括号的先做指数运算.

(2).先乘除后加减,负指数幂化成正指数幂的倒数.

(3).底数是负数,先确定符号;底数是小数,先化成分数;底数是带分数的,先化成假分数.

(4).若是根式,应化为分数指数幂,尽可能用幂的形式表示,运用指数幂的运算性质来解答.

典例剖析

计算 (0.027^{-frac{1}{3}}-(-cfrac{1}{7})^{-2}+(2cfrac{7}{9})^{frac{1}{2}}-(sqrt{2}-1)^0)

解析:原式=([(0.3)^3]^{-frac{1}{3}}-[(cfrac{1}{7})^{2}]^{-1}+[(cfrac{5}{3})^2]^{frac{1}{2}}-1)

(=cfrac{10}{3}-49+cfrac{5}{3}-1=-45)

化简 (cfrac{sqrt{sqrt[3]{ab^{2} a^{3} b^{2}}}}{sqrt[3]{b}left(a^{frac{1}{6}} b^{frac{1}{2}}right)^{4}}) ((a, b)为正数)的结果是__________.

解:原式=(cfrac{left(left(a b^{2}right)^{frac{1}{3}} cdot a^{3} cdot b^{2}right)^{frac{1}{2}}}{b^{frac{1}{3}} cdot a^{frac{2}{3}} cdot b^{2}}=a^{frac{1}{6}+frac{3}{2}-frac{2}{3}} b^{frac{1}{3}+1-frac{1}{3}-2}=cfrac{a}{b})

计算 (1.5^{-frac{1}{3}}timesleft(-frac{7}{6}right)^{0}+8^{frac{1}{4}} times sqrt[4]{2}+(sqrt[3]{2} times sqrt{3})^{6}-sqrt{left(-frac{2}{3}right)^{frac{2}{3}}})

解析:原式=(left(cfrac{3}{2}right)^{-frac{1}{3}}+2^{frac{3}{4}} times 2^{frac{1}{4}}+2^{2} times 3^{3}-left(cfrac{2}{3}right)^{frac{1}{3}}=left(cfrac{2}{3}right)^{frac{1}{3}}+2+4 times 27-left(cfrac{2}{3}right)^{frac{1}{3}}=110) .

计算 ((0.25)^{frac{1}{2}}-left[-2 timesleft(frac{3}{7}right)^{0}right]^{2} timesleft[(-2)^{3}right]^{frac{4}{3}}+(sqrt{2}-1)^{-1}-2^{frac{1}{2}})

解析:原式=(cfrac{1}{2}-4 times 16+sqrt{2}+1-sqrt{2}=cfrac{1}{2}-64+1=cfrac{1}{2}-63=-cfrac{125}{2})

计算 ((sqrt[3]{2} times sqrt{3})^{6}+(-2018)^{0}-4 timesleft(frac{16}{49}right)^{-frac{1}{2}}+sqrt[4]{(3-pi)^{4}})

解析:原式=(108+1-7+pi-3=99+pi)

计算 (cfrac{a^{frac{3}{2}}-1}{a+a^{frac{1}{2}}+1}-cfrac{a+a^{frac{1}{2}}}{a^{frac{1}{2}}+1}+cfrac{a-1}{a^{frac{1}{2}}-1})

解析:原式=(cfrac{left(a^{frac{1}{2}}-1right) cdotleft(a+a^{frac{1}{2}}+1right)}{a+a^{frac{1}{2}}+1}-cfrac{a^{frac{3}{2}}-a+a-a^{frac{1}{2}}-a^{frac{3}{2}}+a^{frac{1}{2}}-a+1}{a-1})

(=a^{frac{1}{2}}-1-cfrac{1-a}{a-1}=a^{frac{1}{2}})

(x+x^{-1}=3), 求值:(cfrac{x^{frac{3}{2}}+x^{-frac{3}{2}}-3}{x^{2}+x^{-2}-6})

解析:若 (x+x^{-1}=3), 则 (left(x+x^{-1}right)^{2}=9), 即 (x^{2}+x^{-2}=7)

(left(x^{frac{1}{2}}+x^{-frac{1}{2}}right)^{2}=x+2+x^{-1}=5)

且因为 (x+x^{-1}=3>0), 所以 (x>0)(x^{frac{1}{2}}+x^{-frac{1}{2}}=sqrt{5})

(x^{frac{3}{2}}+x^{-frac{3}{2}}=left(x^{frac{1}{2}}+x^{-frac{1}{2}}right)left(x+x^{-1}-1right)=2sqrt{5})

所以 (cfrac{x^{frac{3}{2}}+x^{-frac{3}{2}}-3}{x^{2}+x^{-2}-6}=cfrac{2sqrt{5}-3}{7-6}=2 sqrt{5}-3)


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