Question

Is $\sqrt[4]{75}$ an example of pure surd?

Answer 1

Hint: The knowledge of the number system is a must for solving this problem. There should be no confusion about the concept of surds and indices. A number whose power is in fractional form such that the numerator is smaller than the denominator is known as a surd. So, any number which can be represented as ${{a}^{\dfrac{m}{n}}}$ such that n > m is a proper surd.

Complete step by step answer:
In mathematics, the number system is the branch that deals with various types of numbers possible to form and easy to operate with different operators such as addition, multiplication and so on.
Four major types of numbers can be classified as:
Natural numbers, whole numbers, integers, and rational numbers. Irrational numbers come under the category of imaginary numbers.
Further, numbers are also represented in terms of power such that a raised to power m is expressed as ${{a}^{m}}$. If this power term is integer then the number is not called as surd. But if the power is in fractional form such that the numerator is smaller than the denominator, then it is known as a surd.
This type of representation comes under the family of irrational numbers. Irrational numbers are very clumsy to handle. The reason we leave the number in the form of surd is because in decimal form they would go on forever without a terminating point.
So, in our problem we have to evaluate the number $\sqrt[4]{75}$ as a pure surd or not.
Since, $\sqrt[4]{75}={{\left( 75 \right)}^{\dfrac{1}{4}}}$ cannot be reduced further.
Hence, $\sqrt[4]{75}$ is a pure surd.

Note: The knowledge of the number system and representation of various numbers is essential to solve this problem. Pure surd is a type of surd which is non-reducible and hence has no rational factor except unity. This knowledge is very useful in solving future mathematical problems.