Order of a Surd

The order of a surd indicates the index of root to be extracted.

In \(\sqrt[n]{a}\), n is called the order of the surd and a is called the radicand.

For example: The order of the surd \(\sqrt[5]{z}\) is 5.

(i) A surd with index of root 2 is called a second order surd or quadratic surd.

Example: √2, √5, √10, √a, √m, √x, √(x + 1) are second order surd or quadratic surd (since the indices of roots are 2).

(ii) A surd with index of root 3 is called a third order surd or cubic surd.

Example: ∛2, ∛5, ∛7, ∛15, ∛100, ∛a, ∛m, ∛x, ∛(x - 1) are third order surd or cubic surd (since the indices of roots are 3).

(iv) A surd with index of root 4 is called a fourth order surd.

Example: \(\sqrt[4]{2}\), \(\sqrt[4]{3}\), \(\sqrt[4]{9}\), \(\sqrt[4]{17}\), \(\sqrt[4]{70}\), \(\sqrt[4]{a}\), \(\sqrt[4]{m}\), \(\sqrt[4]{x}\), \(\sqrt[4]{(x - 1}\) are third order surd or cubic surd (since the indices of roots are 4).

(v) In general, a surd with index of root n is called a n\(^{th}\) order surd.

Example: \(\sqrt[n]{2}\), \(\sqrt[n]{3}\), \(\sqrt[n]{9}\), \(\sqrt[n]{17}\), \(\sqrt[n]{70}\), \(\sqrt[n]{a}\), \(\sqrt[n]{m}\), \(\sqrt[n]{x}\), \(\sqrt[n]{(x - 1}\) are nth order surd (since the indices of roots are n).


Problem on finding the order of a surd:

Express ∛4 as a surd of order 12.

Solution:

Now, ∛4

= 4\(^{1/3}\)

= 4\(4^{\frac{1 × 4}{3 × 4}}\), [Since, we are to convert order 3 into 12, so we multiply both numerator and denominator of 1/3 by 4]

= 4\(^{4/12}\)

= \(\sqrt[12]{4^{4}}\)

= \(\sqrt[12]{256}\)














11 and 12 Grade Math

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