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).
(iii) 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).
(iv) 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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