Q. 105.0( 1 Vote )

# In the following equations, find which of the variables x, y, z etc. representrational numbers andĂÂ 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

Answer :

(i) x

^{ 2 }= 5

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)

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