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.

An irrational number  is one that cannot be written as a fraction. All integers are rational numberes.

Repeating decimals are never irrational,  can be eliminated because

.

  and  are perfect squares making them both integers.

Therefore, the only remaining answer is .

Irrational numbers are numbers that can't be expressed as a fracton. This elminates statement III automatically as it's a fraction.

Statement I's fraction is  so this statement is false.

Statement IV. may not be easy to spot but if you let that decimal be  and multiply that by  you will get . This becomes . Subtract it from  and you get an equation of . 

 becomes  which is a fraction.

Statement II can't be expressed as a fraction which makes this the correct answer.

Irrational numbers can't be expressed as a fraction with integer values in the numerator and denominator of the fraction.

Irrational numbers don't have repeating decimals.

Because of that, there is no definite value of irrational numbers.

Therefore,  is irrational because it can't be expressed as a fraction. 

Let's take two irrationals like  and multiply them. The answer is  which is rational.

But what if we took the product of  and . We would get  which doesn't have a definite value and can't be expressed as a fraction.

This makes it irrational and therefore, the answer is sometimes irrational, sometimes rational. 

Some answers can be solved. Let's look at some obvious irrational numbers. 

 is surely irrational as we can't get an exact value.

The same goes for  and . 

 is not a perfect cube so that answer choice is wrong.

Although  is a square root, the sum inside however, makes it a perfect square so that means  is rational. 

Let's look at all the answer choices.

The area of a triangle is base times height divided by two. Since base and height can be any value, this statement is wrong. We can have irrational values or rational values, thus generating both irrational or rational answers. 

The diagonal of a right triangle will generate sometimes rational answers or irrational values. If you have a perfect Pythagorean Triple  or  etc..., then the diagonal is a rational number. A Pythagorean Triple is having all the lengths of a right triangle being rational values. One way the right triangle creates an irrational value is when it's an isosceles right triangle. If both the legs of the triangle are , the hypotenuse is  

, , ,  can't be negative since lengths of triangle aren't negative. 

The same idea goes for volume of cube and area of square. It will generate both irrational and rational values.

The only answer is finding value of .  is irrational and raised to any power except 0 is always irrational. 

The definition of irrational numbers is that they are real numbers that cannot be expressed in a common ratio or fraction.

The term  is imaginary which equals to .

The other answers can either be simplified or can be written in fractions.

The only possible answer shown is .