【指数関数】『指数の計算』指数法則の確認と例題

指数法則

指数法則 (問題)

(1)　$4^5\times 2^{-8}÷8^{-2}$

(2)　$(a^{-1}{)}^3\times a^7÷a^2$

(3)　$(a^2{b}^{-1}{)}^3÷(a{b}^{-2}{)}^2$

(4)　$\sqrt[3]{9}\times \sqrt[3]{81}$

(5)　$\sqrt[3]{5}÷\sqrt[12]{5}\times \sqrt[8]{25}$

(6)　$\sqrt[3]{54}+\sqrt[3]{-250}-\sqrt[3]{-16}$

(7)　${\displaystyle \frac{\sqrt[3]{a^4}}{\sqrt{b}}\times \frac{\sqrt[3]{b}}{\sqrt[3]{a^2}}\times \sqrt[3]{a\sqrt{b}}}$

解説

(1)　$4^5\times 2^{-8}÷8^{-2}$

$=(2^2{)}^5\times {\displaystyle \frac{1}{2^8}÷(2^3{)}^{-2}}$
$=(2^2{)}^5\times {\displaystyle \frac{1}{2^8}÷\frac{1}{2^6}}$
$=2^{10}\times {\displaystyle \frac{1}{2^8}÷\frac{1}{2^6}}$
$={\displaystyle \frac{2^{10}\times 2^6}{2^8}}$
$={\displaystyle \frac{2^{16}}{2^8}}$
$=2^8=256$

(2)　$(a^{-1}{)}^3\times a^7÷a^2$

$=a^{-3}\times a^7\times {\displaystyle \frac{1}{a^2}}$
$={\displaystyle \frac{a^4}{a^2}=a^2}$

(3)　$(a^2{b}^{-1}{)}^3÷(a{b}^{-2}{)}^2$

$=a^6{b}^{-3}÷a^2{b}^{-4}$
$={\displaystyle \frac{a^6}{b^3}\times {\displaystyle \frac{b^4}{a^2}}}$
$={\displaystyle \frac{a^6{b}^4}{a^2{b}^3}=a^{4}b}$

(4)　$\sqrt[3]{9}\times \sqrt[3]{81}$

$=9^{\frac{1}{3}}\times {81}^{\frac{1}{3}}$
$=(3^2{)}^{\frac{1}{3}}\times (3^4{)}^{\frac{1}{3}}$
$=3^{\frac{2}{3}}\times 3^{\frac{4}{3}}$
$=3^{\frac{2}{3}+\frac{4}{3}}$
$=3^2=9$

(5)　$\sqrt[3]{5}÷\sqrt[12]{5}\times \sqrt[8]{25}$

$=5^{\frac{1}{3}}÷5^{\frac{1}{12}}\times (5^2{)}^{\frac{1}{8}}$
$=5^{\frac{1}{3}-\frac{1}{12}}\times 5^{\frac{1}{4}}$
$=5^{\frac{1}{4}}\times 5^{\frac{1}{4}}$
$=5^{\frac{1}{2}}=\sqrt{2}$

(6)　$\sqrt[3]{54}+\sqrt[3]{-250}-\sqrt[3]{-16}$

$={54}^{\frac{1}{3}}+(-250{)}^{\frac{1}{3}}-(-16{)}^{\frac{1}{3}}$
$=(27\cdot 2{)}^{\frac{1}{3}}+(-125\cdot 2{)}^{\frac{1}{3}}-(-8\cdot 2{)}^{\frac{1}{3}}$
$=3\cdot 2^{\frac{1}{3}}-5\cdot 2^{\frac{1}{3}}+2\cdot 2^{\frac{1}{3}}$
$=3\sqrt[3]{2}-5\sqrt[3]{2}+2\sqrt[3]{2}=0$

(7)　${\displaystyle \frac{\sqrt[3]{a^4}}{\sqrt{b}}\times \frac{\sqrt[3]{b}}{\sqrt[3]{a^2}}\times \sqrt[3]{a\sqrt{b}}}$

$={\displaystyle \frac{(a^4{)}^{\frac{1}{3}}}{b^{\frac{1}{2}}}\times \frac{b^{\frac{1}{3}}}{(a^2{)}^{\frac{1}{3}}}\times (a{b}^{\frac{1}{2}}{)}^{\frac{1}{3}}}$
$={\displaystyle \frac{a^{\frac{4}{3}}}{b^{\frac{1}{2}}}\times \frac{b^{\frac{1}{3}}}{a^{\frac{2}{3}}}\times a^{\frac{1}{3}}{b}^{\frac{1}{6}}}$
$={\displaystyle \frac{a^{\frac{4}{3}}\times a^{\frac{1}{3}}}{a^{\frac{2}{3}}}\times \frac{b^{\frac{1}{3}}\times b^{\frac{1}{6}}}{b^{\frac{1}{2}}}}$
$=a^{\frac{4}{3}+\frac{1}{3}-\frac{2}{3}}\times b^{\frac{1}{3}+\frac{1}{6}-\frac{1}{2}}$
$=a^{\frac{4}{3}+\frac{1}{3}-\frac{2}{3}}\times b^{\frac{2}{6}+\frac{1}{6}-\frac{3}{6}}$
$=a^1{b}^0=a$

おわりに