Question

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Answer 1

$1.;frac{sqrt[20]{10}cdotsqrt[5]{10}}{sqrt[4]{10}}=10^{frac1{20}}cdot10^{frac15}cdot10^{-frac14}=10^{frac1{20}+frac15-frac14}=10^{frac1{20}+frac4{20}-frac5{20}}=10^0=1\\2.;frac{sqrt[5]{20}cdotsqrt[5]{8}}{sqrt[5]5}=frac{sqrt[5]{20cdot8}}{sqrt[5]5}=sqrt[5]{frac{160}5}=sqrt[5]{32}=2\\3.;8cdotsqrt[3]{81}cdotsqrt[6]{81}=8cdot81^{frac13}cdot81^{frac16}=8cdot81^{frac13+frac16}=8cdot81^{frac26+frac16}=8cdot81^{frac12}=8sqrt{81}=8cdot9=72$

$4.;left(sqrt{3frac67}-sqrt{1frac57}right):sqrt{frac3{175}}=left(sqrt{frac{27}7}-sqrt{frac{12}7}right)cdotsqrt{frac{175}3}=sqrt{frac{27}7}cdotsqrt{frac{175}3}-sqrt{frac{12}7}cdotsqrt{frac{175}3}=\=sqrt{frac{27}7cdotfrac{175}3}-sqrt{frac{12}7cdotfrac{175}3}=sqrt{9cdot25}-sqrt{4cdot25}=(sqrt9-sqrt4)cdotsqrt{25}=(3-2)cdot5=5$

$left(sqrt{15}-sqrt{14}right)(sqrt{15}+sqrt{14})=left(sqrt{15}right)^2-left(sqrt{14}right)^2=15-14=1$