指数幂的运算

前言

运算法则

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

公式:\(a^m\cdot a^n=a^{m+n}\)\((a^m)^n=(a^n)^m=a^{mn}\)\((a\cdot b)^n=a^n\cdot b^n\)\((\cfrac{a}{b})^n=\cfrac{a^n}{b^n}=a^n\cdot b^{-n}\)

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

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

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

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

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=\(\cfrac{\left(a^{\frac{1}{2}}-1\right) \cdot\left(a+a^{\frac{1}{2}}+1\right)}{a+a^{\frac{1}{2}}+1}-\cfrac{a^{\frac{3}{2}}-a+a-a^{\frac{1}{2}}-a^{\frac{3}{2}}+a^{\frac{1}{2}}-a+1}{a-1}\)

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

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

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

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

指数幂的运算

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