In the following equations, find which of the variables x, y, z etc. representrational numbers and which represent irrational numbers:
(i) x ² = 5
(ii)y ² = 9
(iii) z ² = 0.04
(iv) u ² = 17/4
(v) v ² = 3
(vi) w ³ = 27
(vii) t ² = 0.4

Taking square root both sides
∴ x=√5

It cannot be expressed as ratio of two integers and value of √5= 2.2360679… which is non termination non repeating.

Hence, x is an irrational number.

Decimal representation of x =2.2360679…

(ii) y ² = 9

Taking square root both sides

∴ y=√9

∴ y = 3 =3/1

It can be represented in form p/q (p=3, q=1) where q≠0 and p;q have no common factor other than 1.

Hence, y is a rational number.

(iii) z ² = 0.04

Taking square root both sides

z=√0.04 = √0.2×0.2
∴ z = 0.2=2/10

z= 1/5

It can be represented in form p/q (p=1, q=5) where q≠0 and p;q have no common factor other than 1.

Hence, z is a rational number.

Decimal representation of z is 0.2.

(iv)

Taking square root both sides

It cannot be expressed as ratio of two integers and value of √17= 4.12310562… which is non termination non repeating.

Hence, u is an irrational number.
Decimal representation of u is 2.06155281… (4.12310562…/2)

(v) v ² = 3

Taking square root both sides

v=√3

It cannot be expressed as ratio of two integers and value of √3 = 1.7320508075688… which is non termination non repeating.

Hence, v is an irrational number.

Decimal representation of v is 1.7320508075688…

(vi) w ³ = 27

Taking cube root both sides

∴ w=3  =3/1

It can be represented in form p/q (p=3, q=1) where q≠0 and p;q have no common factor other than 1.

Hence, w is a rational number.

(vii) t ² = 0.4

Taking square root both sides

It cannot be expressed as ratio of two integers and value of √10 = 3.16227766… which is non termination non repeating.

Hence, t is an irrational number.

Decimal representation of t is 0.6324555…(2/√10)