Real numbers may include positive and negative integers, rational numbers, and also irrational numbers.  

Rational numbers are terms that can be expressed as ratios of numbers, such as $\displaystyle \frac{3}{7}$.

Real numbers cannot include infinity or imaginary numbers, such as $\displaystyle i=\sqrt{-1}$.

However, the imaginary term provided in the answer choice is not fully simplified.

$\displaystyle i^4 = i^2\cdot i^2 = (-1)(-1) = 1$

This value is a real number.

The term $\displaystyle \frac{\pi}{e}$ is an irrational number, and is considered a real number.  

The term $\displaystyle \sqrt{30}$ is also irrational, and is a real number.

The correct answer is:  $\displaystyle \textup{All of the answers are real numbers.}$

$\displaystyle \pi$ is a real number, because you can represent it on the Cartesian coordinate plane, but it is irrational because it cannot be represented by a fraction of two integers. Natural numbers are integers greater than 0.

Natural numbers are simply whole, non-negative numbers. 

Using this definition, we see only one set of numbers within our answer choices containing only whole, non-negative numbers. Any set containing decimals or negative numbers, will violate our defintion of natural numbers and thus be an incorrect answer. 

There is a repeating pattern of four exponent values of $\displaystyle i: i^1=i, i^2=-1, i^3=-1, i^4=1$.

$\displaystyle i^6$ is the same as $\displaystyle i^2$.

$\displaystyle i^6=i^2\cdot i^2 \cdot i^2=(-1)(-1)(-1)=-1$

Multiplying out using FOIL (First, Inner, Outer, Last) results in,

 $\displaystyle 2i^2-8i+2i-8$.

Remember that $\displaystyle i^2=-1$  

$\displaystyle 2(-1)-8i+2i-8=-6i-10$

For a number to be prime it must only have factors of one and itself.

10 has factors 1, 2, 5, 10.

15 has factors 1, 3, 5, 15.

18 has factors 1, 2, 3, 6, 9, 18.

The only factors of 13 are 1 and 13. As such it is prime.

Irrational numbers are defined by the fact that they cannot be written as a fraction which means that the decimals continue forever. 

Looking at our possible answer choices we see,

$\displaystyle 1.7392=\frac{1087}{625}$

$\displaystyle \frac{4}{5}$ is already in fraction form

$\displaystyle 3-2i=3-2\sqrt{-1}$ which is an imaginary number but still rational.

Therefore since,

 $\displaystyle \sqrt{2} = 1.414213562......$

we can conclude it is irrational.

An irrational number is a number that cannot be written in fraction form. In other words a nonrepeating decimal is an irrational number.

The $\displaystyle \sqrt{3}$ is an irrational number. 

$\displaystyle i^2$ is a real number with a value of $\displaystyle -1$.

Therefore, $\displaystyle x=-\sqrt{3}$. This is a real but irrational number.

The graph shown never intersects with the x-axis. This means that the x-intercepts must be imaginary.

The square root of 5 is irrational since it is a non-terminating, non-repeating decimal that cannot be expressed as a fraction.