Classify the given pair of surds into like surds and unlike surds.

i. √52, 5√13

ii. √68, 5√3

iii. 4√18, 7√2

iv. 19√12, 6√3

v. 5√22, 7√33

vi. 5√5, √75

Two or more surds are said to be similar or like surds if they have the same surd-factor.

And,

Two or more surds are said to be dissimilar or unlike when they are not similar.

Therefore,

i. √52, 5√13

√52 = √(2×2×13) = 2√13

5√13

∵ both surds have same surd-factor i.e., √13.

∴ they are like surds.

ii. √68, 5√3

√68 = √(2×2×17) = 2√17

5√3

∵ both surds have different surd-factors √17 and √3.

∴ they are unlike surds.

iii. 4√18, 7√2

4√18 = 4√(2×3×3) = 4×3√2 = 12√2

7√2

∵ both surds have same surd-factor i.e., √2.

∴ they are like surds.

iv. 19√12, 6√3

19√12 = 19√(2×2×3) = 19×2√3 = 38√3

6√3

∵ both surds have same surd-factor i.e., √3.

∴ they are like surds.

v. 5√22, 7√33

∵ both surds have different surd-factors √22 and √33.

∴ they are unlike surds.

vi. 5√5, √75

5√5

√75 = √(5×5×3) = 5√3

∵ both surds have different surd-factors √5 and √3.

∴ they are unlike surds.