Answer:
[tex]1. Solution,\\Let~x=0.272727......\\Then, 100x = 27.272727.....\\Now,\\100x-x = 27.272727..... - 0.27272727..... = 27.\\or, 99x = 27\\or, x = \frac{27}{99} = \frac{3}{11}\\Thus,~\frac{3}{11} =0.27272727......\\[/tex]
[tex]2. (27)^\frac{2}{3}.(125)^\frac{1}{3}\\= (3^3)^\frac{2}{3}.(5^3)^\frac{1}{3}\\=(3^2)(5)\\=45.[/tex]
[tex]3. Solution,\\Here, \frac{1}{5} = 0.2\\\frac{1}{2} = 0.5\\Now, ~Irrational~ numbers~ between~ 0.2 ~and~ 0.5 ~are~ 0.2122275647.......~and~0.467858930.......\\[/tex]
You can write any numbers between 0.2 and 0.5 which follow irregular patterns like this.
[tex]OR\\Examples~of~such~irrational~numbers~are~5+\sqrt{7}~ and~5-\sqrt{7} .[/tex]
[tex]4. Solution,\\x^2+y^2 = (x+y)^2-2xy=5^2-2(10) = 25-20=5.\\5.~ If~x=-3~and~y=-2~are~the~solutions~of~the~equation~2x-3y=k,~then,\\2(-3)-3(-2)=k\\or, k = -6+6\\or, k = 0[/tex]
