Answer 1

We are asked to represent `sqrt6,` `sqrt7` and `sqrt8`on the number line

We will follow certain algorithm to represent these numbers on real line

We will consider point A as reference point to measure the distance

(1) First of all draw a line AX and YY’ perpendicular to AX

(2) Consider AO = 2 unit and OB = 1 unit, so

`AB=sqrt(2^2+1^2)=sqrt5`

(3) Take A as center and AB as radius, draw an arc which cuts line AX at A₁

(4) Draw a perpendicular line A₁B₁ to AX such that A₁B₁ = 1 unit and 

(5) Take A as center and AB₁ as radius and draw an arc which cuts the line AX at A₂.

Here 

AB₁ = AA₂

`=sqrt("AA"_1^2+A_1B_1^2)`

`=sqrt((sqrt5)^2+1)`

`=sqrt6` unit

So AA₂ = `sqrt6` unit

So A₂ is the representation for `sqrt6`

(1) Draw line A₂B₂ perpendicular to AX

(2) Take A center and AB₂ as radius and draw an arc which cuts the horizontal line at A₃ such that 

AB₂ = AA₃

`=sqrt("AA"_2^2+A_2B_2^2)`

`=sqrt((sqrt6)^2+1)`

`=sqrt7` unit

So point A₃ is the representation of `sqrt7`

(3) Again draw the perpendicular line A₃B₃ to AX

(4) Take A as center and AB₃ as radius and draw an arc which cuts the horizontal line at A₄ 

Here;

AB₃ = AA₄

`=sqrt("AA"_3^2+A_3B_3^2)`

`=sqrt((sqrt7)^2+1^2)`

`=sqrt8` unit

A₄ is basically the representation of `sqrt8`