An irrational number is defined as any number that cannot be expressed as a simple fraction or does not have terminating or repeating decimals. Of the answer choices given, the only number that cannot be expressed as a simple fraction or with repeating or terminating decimals is .

An irrational number is any number that can not be expressed as a ratio of integers, i.e. a fraction. Therefore, the only irrational number listed is .

The square root of an integer is either an irrational number or an integer. The latter is the case if and only if there is an integer which, when multiplied by itself, or squared, yields the number inside the symbol (the radicand) as the product. Of , only 81 is the square of an integer (9).

Pi is the only irrational number listed. Irrational numbers are in the form of infinite non-repeating decimals. 

A root of an integer is one of two things, an integer or an irrational number. By testing all five on a calculator, only  comes up an exact integer - 5. This is the correct choice.

An irrational number is any number that cannot be written as a fraction of whole numbers.  The number pi and square roots of non-perfect squares are examples of irrational numbers.  

 can be written as the fraction .  The term  is a whole number.  The square root of  is , also a rational number. , however, is not a perfect square, and its square root, therefore, is irrational.

A rational number is any number that can be expressed as a fraction/ratio, with both the numerator and denominator being integers. The one limitation to this definition is that the denominator cannot be equal to .

Using the above definition, we see  ,  and   (which is ) cannot be expressed as fractions. These are non-terminating numbers that are not repeating, meaning the decimal has no pattern and constantly changes. When a decimal is non-terminating and constantly changes, it cannot be expressed as a fraction.

  is the correct answer because , which can be expressed as , fullfilling our above defintion of a rational number. 

The definition of an irrational number is a number which cannot be expressed in a simple fraction, or a number that is not rational.

Using the above definition, we see that  is already expressed as a simple fraction.

  any number  and

. All of these options can be expressed as simple fractions, making them all rational numbers, and the incorrect answers.

  cannot be expressed as a simple fraction and is equal to a non-terminating, non-repeating (ever-changing) decimal, begining with 

This is an irrational number and our correct answer.

Rational numbers are those which can be written as a ratio of two integers, or simply, as a fraction.

The solution of  is , which can be written as . Each of the other answers would have a solution with an infinite number of decimal points, and therefore cannot be written as a simple ratio. They are irrational numbers. 

An irrational number cannot be represented as the quotient of two integers.

Irrational numbers do not terminate and are not repeat numbers.

Looking at the possible answers,

 can be reduced to , therefore it is an integer.

 by definition is a quotient of two integers and thus it is not an irrational number.

 can be rewritten as  and by definition is a quotient of two integers and thus it is not an irrational number.

 is a terminated decimal and therefore can be written as a fraction. Thus it is not an irrational number.

 is the number for  and does not terminate, therefore it is irrational.