**Similar and Dissimilar Surds**

A root of a positive real quantity is called a surd if its value cannot be exactly determined. It is a number that can’t be simplified to remove a square root (or cube root etc). For example, each of the quantities √3, ∛7, ∜19, (16)^{2/5} etc. is a surd.

More Examples:

- √2 (square root of 2) can’t be simplified further so it is a surd
- √4 (square root of 4) CAN be simplified to 2, so it is NOT a surd

**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√(5∙5∙5) = 2√5 and 5^{5/2} = √5^{5} = √(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 = ^{n}√4^{n}

In general, if a he a rational number then,

x = √x^{2} = ∛x^{3} = ∜x^{4} = ^{n}√x^{2}.

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