# Similar Surds and Dissimilar Surds

We have learned about surds, different types of surds, rules of surds, etc on the page an introduction to surd. In this section, we will focus on what are the similar or like surds and what are the dissimilar or unlike surds.

#### Definition of Similar Surds:

Two or more surds are called similar surds if

(i) they have the same surd-factor, or

(ii) they can be reduced to the surds with the same surd-factor.

Note that similar surds are also known as like surds. From the above definition, it is clear that:

Definition of like surds: Two surds (or more surds) are called similar or like surds, if the surds or their reduced forms have the same surd-factor.

#### Examples of Similar or like Surds:

(i) $\sqrt{2},$ $-3\sqrt{2},$ $7\sqrt{2}$ are examples of similar surds as they have the same surd-factor $\sqrt{2}.$

(ii) Similarly, $\sqrt{3},$ $5\sqrt{3}$ are examples of like surds.

(iii) Now consider the surds $3\sqrt{2},$ $2^{5/2},$ $\sqrt{8}.$ Note that the second surd $2^5/2$ $=(2^5)^{1/2}$ $=\sqrt{32}$ $\sqrt{16 \times 2}$ $=\sqrt{16} \times \sqrt{2}$ $=4\sqrt{2}$ and the third surd $\sqrt{8}$ $=\sqrt{4 \times 2}$ $=\sqrt{4} \times \sqrt{2}$ $=2\sqrt{2}.$ Thus the given surds can be reduced to the surds with surd-factor $\sqrt{2}.$ So by the definition, we conclude that the given surds $3\sqrt{2},$ $2^{5/2},$ $\sqrt{8}$ are similar.

(iv) $2\sqrt{2},$ $3\sqrt{3},$ $5\sqrt{5},$ $7\sqrt{2}$ are not similar surds. They are not like surds as they have different surd-factors $\sqrt{2},$ $\sqrt{3}$ and $\sqrt{5}.$

Now we will discuss about dissimilar or unlike surds.

#### Definition of Dissimilar Surds:

If two or more surds are not similar, then they are called dissimilar surds. These surds are also known as unlike surds. More precisely, two or more surds are said to be unlike or dissimilar if

(i) they do not have the same surd-factor, or

(ii) they cannot be reduced to the surds having the same surd-factor.

#### Examples of Dissimilar or unlike Surds:

(i) $\sqrt{2},$ $\sqrt{3}$ are unlike surds.

(ii) Consider the surds $2^{3/2},$ $\sqrt{2},$ $3^{3/2}.$ Note that the first surd $2^{3/2}$ $=\sqrt{2^3}$ $=\sqrt{2 \times 2 \times 2}$ $=2\sqrt{2}$ and the last surd $3^{3/2}$ $=\sqrt{3^3}$ $\sqrt{3 \times 3 \times 3}$ $=3\sqrt{3}.$ Thus $2^{3/2},$ $\sqrt{2},$ $3^{3/2}$ have the surd-factors $\sqrt{2}$ and $\sqrt{3}.$ As they have different surd-factors, the given surds are dissimilar surds or unlike surds.

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