Question

State which of the following are pure surds and mixed surds:
\[\dfrac{3}{4}\sqrt 8 \]
A. Pure surd
B. Mixed surd
C. Not a surd
D. None of above

Answer 1

Hint: We solve this using the information about surds that any number which cannot be simplified to remove it’s radical then it is called a surd and the classification of surds which is
Pure surd: A surd having no rational factor except unity is called a pure surd, i.e. they have rational numbers under the radical sign only. Examples: \[\sqrt 3 ,\sqrt {15} ,\sqrt[3]{{13}}\] etc.
Mixed surd: A surd having rational factor other than unity is called a mixed surd, i.e. some part of the quantity inside the radical is taken out of the radical. Examples: \[2\sqrt 3 ,5\sqrt {15} ,7\sqrt[3]{{13}}\]etc.
* A surd always has its decimal representation as non-terminating and non-repeating which means the digits after the decimal are never ending and do not repeat to form approximation to any number.
* Every irrational number (a number whose under root cannot be simplified) is a surd.

Complete step-by-step answer:
We are given the number \[\dfrac{3}{4}\sqrt 8 \]
We check if it is a mixed surd or a pure surd from the definitions of mixed and pure surds that there is factor unity only outside the radical then it is pure surd and if there is factor other than unity outside the radical then it is mixed surd.
Since there is factor other than unity along with the radical, i.e. \[\dfrac{3}{4}\]
So, the number is a mixed surd.

So, the correct answer is “Option b”.

Note: Students are likely to get confused with the fraction outside the radical but it is also a factor other than unity which is good enough to decide which surd is the number.