State with reason which of the following are surds and which are not?
(i) √5×√10
(ii) √8×√6
(iii) √27×√3
(iv) √16×√4
(v) 5√8×2√6
(vi)√125×√5
(vii) √100×√2
(viii) 6√2×9√3
(ix)√120×√45
(x) √15×√6

An irrational root of a rational number is a surd.

(i) √5×√10  =√5×√(5×2)= √5×√5×√2=5×√2

5√2  is an irrational number

Since, it cannot be expressed as ratio of two integers and value of √2 =1.414213562… which is non termination non repeating.

Thus √5×√10 is a surd.

(ii) √8×√6= √(4×2)×√(3×2)=√4 √2 √2 √3=2×2×√3=4√3

This is an irrational number.

Since, it cannot be expressed as ratio of two integers and value of √3=1.73205080… which is non termination non repeating.

Thus √8×√6  is a surd

(iii) √27×√3= √(9×3)×√3=3×√3×√3=3×3=9 which is a rational number. (9/1)

Since it can be represented in form p/q (p=9, q=1) where q≠0 and p;q have no common factor other than 1.

Thus √27×√3  is a not a surd.

(iv) √16×√4=4×2=8/1 which is a rational number.

Since, it can be represented in form p/q (p=8, q=1) where q≠0 and p;q have no common factor other than 1.

Thus √16×√4  is not a surd.

(v) 5√8×2√6=5√(4×2)×2√(3×2)=5×√4×√2×2×√3×√2

5×2×√2×2×√3×√2=40√3  is irrational

Since, it cannot be expressed as ratio of two integers and value of √3=1.73205080… which is non termination non repeating.

Thus 5√8×2√6  is a surd.

(vi) √125×√5= √(5×5×5)×√5= √(5×5)  × √(5×5)=5×5=25/1 is a rational number

Since, it can be represented in form p/q (p=25, q=1) where q≠0 and p;q have no common factor other than 1.

Thus √125×√5  is not a surd.

(vii) √100×√2=10√2  is an irrational number

Since, it cannot be expressed as ratio of two integers and value of √2 =1.414213562… which is non termination non repeating.

Thus √100×√2  is a surd.

(viii) 6√2×9√3= 54√2×√3= 54√6  is an irrational number

Since, it cannot be expressed as ratio of two integers and value of √6 =2.449489742… which is non termination non repeating.

Thus 54√2  is a surd.

(ix) √120×√45= √(15×8)×√(15×3)=√15×√15×√(4×2)×√3

=15×2×√2×√3=30√6  is an irrational number

Since, it cannot be expressed as ratio of two integers and value of √6 =2.449489742… which is non termination non repeating.

Thus √120×√45  is a surd.

(x) √15×√6=√(5×3)×√(2×3)= √5×√2×√3×√3=3√10  is an irrational number

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

Thus √15×√6  is a surd.