Question

1.设x=1+2^p,y=1+2^-p,用X的式子表示Y.2.已知5^x=18,5y=3,求25^x-y的值.

Answer 1

(1)

x=1+2^p

2^p=x-1

2^(-p)=1/2^p=1/(x-1)

y=1+1/(x-1)=x/(x-1)

(2)

5^(2x)=(5^x)^2=18^2=324

5^(2y)=(5^y)^2=3^2=9

25^(x-y)=5^(2x-2y)=5^2x/5^2y=324/9=36

(3)

x(x-5)+y(y+6)=x^2+y^2-39

x^2-5x+y^2+6y=x^2+y^2-39

5x-6y=39

x(x+7)-y(y-8)=x^2-y^2-11

x^2+7x-y^2+8y=x^2-y^2-11

7x+8y=-11

5x-6y=39

解得

x=3

y=-4

(4)

3^2001-4*3^2000+10*3^1999

=9*3^1999-12*3^1999+10*3^1999

=3^1999*(9-12+10)

=7*3^1999

所以3^2001-4*3^2000+10*3^1999能被7整除