Similar and Dissimilar Surds

We will discuss about similar and dissimilar surds and their definitions.

Definition of Similar Surds:

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

or,

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

For example,

(i) √5, 7√5, 10√5, -3√5, 5$$^{1/2}$$, 10 ∙ √5, 12 ∙ 5$$^{1/2}$$ are similar surds;

(ii) 7√5, 2√125, 5$$^{2/5}$$are similar surds since 2√125 = 2 ∙ $$\sqrt{5 ∙ 5 ∙ 5}$$ = 2√5 and 5$$^{5/2}$$ =$$\sqrt{5^{5}}$$ = $$\sqrt{5 ∙ 5 ∙ 5 ∙ 5 ∙ 5}$$ = 25√5 i.e., each of the given surds can be expressed with the same surd-factor √5.

Definition of Dissimilar Surds:

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

For example, √2, 9√3, 8√5, ∛6, 17, 7$$^{5/6}$$ are unlike surds.

Note: A given rational number can be expressed in the form of a surd of any desired order.

For example, 4 = √16 = ∛64 = ∜256 = $$\sqrt[n]{4^{n}}$$

In general, if a he a rational number then,

x = √x$$^{2}$$ = ∛x$$^{3}$$ = ∜x$$^{4}$$ = $$\sqrt[n]{x^{n}}$$.