Question 767041
1. If the rectangular piece of paper looks like this: {{{drawing(300,200,-3,33,-2,22,
rectangle(0,0,30,20),line(15,0,15,4),
line(15,8,15,12),line(15,16,15,20),
locate(0.5,11,y),locate(7,2,x),locate(22,2,x)
)}}},
the ratio of the sides is {{{x/y=y/2x}}}, so
{{{x/y=y/2x}}} --> {{{2x^2=y^2}}} --> {{{y=xsqrt(2)}}}
and the area of the rectangle is
{{{(2x)y=100}}} --> {{{(2x)(xsqrt(2))=100}}} --> {{{2sqrt(2)x^2=100}}} --> {{{8x^4=10000}}} --> {{{x^4=1250}}} --> {{{x^4=5^4*2}}} --> {{{x=5root(4,2)}}}
So {{{y=xsqrt(2)}}} --> {{{y=5root(4,2)sqrt(2)}}} --> {{{y=5root(4,2)root(4,4)}}} --> {{{y=5root(4,8)}}}
 
2. {{{(2^(a+3) - 4*2^a)/ (2^(2a+1) - 4^a)=(2^a*2^3 - 4*2^a)/ (2^(2a)*2 - (2^2)^a)=
2^a*(8-4)/ (2^(2a)*2 - 2^(2a))=2^a*(8-4)/ (2^(2a)*(2-1))=4*2^a/ 2^(2a)=2^(a+2)/2^(2a)=1/2^(a-2)}}}
 
3. {{{ sqrt(2)/ (2sqrt(2) + 1) + 2/(sqrt(3) +1) =
sqrt(2)(2sqrt(2) - 1)/ (2sqrt(2) + 1)/(2sqrt(2) - 1) + 2(sqrt(3) -1)/(sqrt(3) +1)/(sqrt(3) -1)=(sqrt(2)*2sqrt(2)-sqrt(2))/((2sqrt(2))^2-1^2)+2(sqrt(3) -1)/((sqrt(3))^2-1^2)
=(8-sqrt(2))/(8-1)+2(sqrt(3) -1)/(3-1)=(8-sqrt(2))/7+2(sqrt(3) -1)/2
=8/7-sqrt(2)/7+sqrt(3) -1=1/7-sqrt(2)/7+sqrt(3)}}}