Задача 30580 9^(log3(3+1/sqrt(2))) ......

89648013987

2018-10-28 10:27:27

9^(log3(3+1/sqrt(2))) ...

sova

2018-10-28 17:19:09

★

9^(log_(3)(3+(1/sqrt(2)))=(3^2)^(log_(3)(3+(1/sqrt(2)))=3^(2*log_(3)(3+(1/sqrt(2)))=3^(log_(3)(3+(1/sqrt(2))^2=(3+(1/sqrt(2))^2=
=9+2*3*(1/sqrt(2))+(1/2)=9,5+3sqrt(2);

25^(log_(1/5)(sqrt(2))/(3sqrt(2)-1))=(5^(2))^(log_(5^(-1))(sqrt(2))/(3sqrt(2)-1))=
=5^(-2log_(5) (sqrt(2)/(3sqrt(2)-1)))=5^(log_(5) (sqrt(2)/(3sqrt(2)-1)^(-2)))=
=((3sqrt(2)-1)/sqrt(2))^2=(3-(1/sqrt(2))^2=9-2*3*(1/sqrt(2))+(1/2)=
=9,5-3sqrt(2)

3)
9,5+3sqrt(2)-(9,5-3sqrt(2))=6sqrt(2);

4) 3^(log_(5)10)=3^(log_(5)2*5)=3^(log_(5)2+log_(5)5)=
=3^(log_(5)2+1)=3^(log_(5)2)*3^(1)=3^(log_(5)2)*3

4)2^(log_(5)3sqrt(5))=2^(log_(5)3+log_(5)sqrt(5))=
=2^(log_(5)3+(1/2)log_(5)5)=2^(log_(5)3)*2^(1/2)=
=sqrt(2)*2^(log_(5)3)

Так как
2^(log_(5)3)=3^(log_(5)2),в чем можно убедиться прологарифмировав равенство по основанию 5 и применив свойство логарифма степени, то

О т в е т. 6sqrt(2)*sqrt(2)/3=4