We will discuss about the pure and mixed surds.
Definition of Pure Surd:
A surd in which the whole of the rational number is under the radical sign and makes the radicand, is called pure surd.
In other words a surd having no rational factor except unity is called a pure surd or complete surd.
For example, each of the surds √7, √10, √x, ∛50, ∛x, ∜6, ∜15, ∜x, 17\(^{2/3}\), 59\(^{5/7}\), m\(^{2/13}\) is pure surd.
Definition of Mixed Surd:
A surd having a rational coefficient other than unity is called a mixed surd.
In other words if some part of the quantity under the radical sign is taken out of it, then it makes the mixed surd.
For example, each of the surds 2√7, 3√6, a√b, 2√x, 5∛3, x∛y, 5 ∙ 7\(^{2/3}\) are mixed surd.
More examples:
√45 = \(\sqrt{3\cdot 3\cdot 5}\) = 3√5 is a mixed surd.
√32 = \(\sqrt{2\cdot 2\cdot 2\cdot 2\cdot 2}\) = 2 × 2 × √2 = 4√2 is a mixed surd.
\(\sqrt[4]{162}\) = \(\sqrt[4]{ 2\cdot 3\cdot 3\cdot 3\cdot 3}\) = 3\(\sqrt[4]{2}\) is a mixed surd.
Note:
I. A mixed surd can be expressed in the form of a pure surd.
For example,
(i) 3√5 = \(\sqrt{3^{2}\cdot 5}\) = \(\sqrt{9 \cdot 5}\) = √45
(ii) 4 ∙ ∛3 = \(\sqrt[3]{4^{3}}\) ∙ ∛3 = \(\sqrt[3]{64}\) ∙ ∛3 = \(\sqrt[3]{64}\cdot 3\) = ∛192
In general, x \(\sqrt[n]{y}\) = \(\sqrt[n]{x^{n}}\) ∙ \(\sqrt[n]{y}\) = \(\sqrt[n]{x^{n}y}\)
II. Sometimes a given pure surd can be expressed in the form of a mixed surd.
For example,
(i) √375 = \(\sqrt{5^{3}\cdot 3}\) = 5√15;
(ii) ∛81 = \(\sqrt[3]{3^{4}}\) = 3∛3
(iii) ∜64 = \(\sqrt[4]{2^{6}}\) = 2\(\sqrt[4]{2^{2}}\)= 2\(\sqrt[4]{4}\)
But ∛20 can't be expressed in the form of mixed surd.
11 and 12 Grade Math
From Pure and Mixed Surds to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.