Сравнить числа:

$1) 0,{88}^{\frac{1}{6}}$

и

${\left(\frac{6}{11}\right)}^{\frac{1}{6}}.$

Если

$0<x_1<x_2$

и

$p>0,$

то

${x_1}^p<{x_2}^p.$ $0,{88}^{\frac{1}{6}}> {\left(\frac{6}{11}\right)}^{\frac{1}{6}}$ $0,88> \frac{6}{11}$ $\frac{88}{100}>\frac{6}{11}$ $\frac{22}{25}>\frac{6}{11}$ $\frac{22·11}{25·11}>\frac{6·25}{11·25}$ $\frac{242}{275}>\frac{150}{275}$ $242>150$ $2) {\left(\frac{5}{12}\right)}^{—\frac{1}{4}}$

и

$0,{41}^{—\frac{1}{4}}.$

Если

$0<x_1<x_2$

и

$p<0,$

то

${x_1}^p>{x_2}^p.$ ${\left(\frac{5}{12}\right)}^{—\frac{1}{4}}<0,{41}^{—\frac{1}{4}}$ $\frac{5}{12}> 0,41$ $\frac{5}{12}>\frac{41}{100}$ $\frac{5·100}{12·100}>\frac{41·12}{100·12}$ $\frac{500}{1200}>\frac{492}{1200}$ $500>492$ $3) 4,{09}^{\sqrt[3]{2}}$

и

${\left(4\frac{3}{25}\right)}^{\sqrt[3]{2}}.$

Если

$0<x_1<x_2$

и

$p>0,$

то

${x_1}^p<{x_2}^p.$ $4,{09}^{\sqrt[3]{2}}<{\left(4\frac{3}{25}\right)}^{\sqrt[3]{2}}$ $4,{09}^{\sqrt[3]{2}}<{\left(\frac{4·25+3}{25}\right)}^{\sqrt[3]{2}}$ $4,{09}^{\sqrt[3]{2}}<{\left(\frac{103}{25}\right)}^{\sqrt[3]{2}}$ $4,{09}^{\sqrt[3]{2}}<{\left(4,12\right)}^{\sqrt[3]{2}}$ $4,09<4,12$ $4) {\left(\frac{11}{12}\right)}^{—\sqrt{5}}$

и

${\left(\frac{12}{13}\right)}^{—\sqrt{5}}.$

Если

$0<x_1<x_2$

и

$p<0,$

то

${x_1}^p>{x_2}^p.$ ${\left(\frac{11}{12}\right)}^{—\sqrt{5}}>{\left(\frac{12}{13}\right)}^{—\sqrt{5}}$ $\frac{11}{12}<\frac{12}{13}$ $\frac{11·13}{12·13}<\frac{12·12}{13·12}$ $\frac{143}{156}<\frac{144}{156}$ $143<144$