\(\displaystyle \sqrt{77 - 17 + 14}\)

\(\displaystyle = \sqrt{60 + 14}\)

\(\displaystyle = \sqrt{74}\)

Since \(\displaystyle 64 < 74 < 81\), 

\(\displaystyle 8 < \sqrt{74} < 9\).

We can determine which is closer by evaluating \(\displaystyle 8.5 ^{2} = 8.5 \times 8.5 = 72.25\).

Since \(\displaystyle 74 > 72.25\), 9 is the closer integer, and it is the correct choice.

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 \(\displaystyle \sqrt[3]{125}\) comes up an exact integer - 5. This is the correct choice.

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

Repeating decimals are never irrational, \(\displaystyle 0.1666...\) can be eliminated because

\(\displaystyle 0.1666...=\frac{1}{6}\).

 \(\displaystyle \sqrt{36}\) and \(\displaystyle \sqrt{196}\) are perfect squares making them both integers.

\(\displaystyle \sqrt36=6, \sqrt196=14\)

Therefore, the only remaining answer is \(\displaystyle \sqrt{278}\).

A rational number can be expressed as a fraction of integers, while an irrational number cannot.  

\(\displaystyle 0.\overline{3333}\) can be written as \(\displaystyle \frac{1}{3}\).   

\(\displaystyle \sqrt{9}\) is simply \(\displaystyle 3\), which is a rational number.  

The number \(\displaystyle 5.75\) can be rewritten as a fraction of whole numbers, \(\displaystyle \frac{23}{4}\), which makes it a rational number.  

\(\displaystyle \frac{21}{5}\)is also a rational number because it is a ratio of whole numbers.  

The number, \(\displaystyle \pi\), on the other hand, is irrational, since it has an irregular sequence of numbers (\(\displaystyle 3.14159\)...) that cannot be written as a fraction.

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.  

\(\displaystyle 3.25\) can be written as the fraction \(\displaystyle \frac{13}{4}\).  The term \(\displaystyle \sqrt{\frac{175}{7}} = \sqrt{25} = 5\) is a whole number.  The square root of \(\displaystyle 10,000\) is \(\displaystyle 100\), also a rational number. \(\displaystyle 79\), 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 \(\displaystyle 0\).

Using the above definition, we see \(\displaystyle \sqrt{2}\) , \(\displaystyle .2020020002...\) and  \(\displaystyle \pi\) (which is \(\displaystyle 3.145926...\)) 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.

\(\displaystyle \sqrt{4}\)  is the correct answer because \(\displaystyle \sqrt{4}=2\), which can be expressed as \(\displaystyle \frac{2}{1}\), 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 \(\displaystyle \frac{1}{2}\) is already expressed as a simple fraction.

\(\displaystyle -1.5 = -\frac{3}{2}\) 

 \(\displaystyle 0\div\) any number \(\displaystyle = 0\) and

\(\displaystyle 3 = \frac{3}{1}\). All of these options can be expressed as simple fractions, making them all rational numbers, and the incorrect answers.

\(\displaystyle \sqrt{5}\)  cannot be expressed as a simple fraction and is equal to a non-terminating, non-repeating (ever-changing) decimal, begining with \(\displaystyle 2.2360679775....\)

This is an irrational number and our correct answer.

Let's take two irrationals like \(\displaystyle \sqrt{3}\) and multiply them. The answer is \(\displaystyle 3\) which is rational.

\(\displaystyle \sqrt3 \cdot \sqrt3=\sqrt{3\cdot 3}=\sqrt9=3\)

But what if we took the product of \(\displaystyle \pi\) and \(\displaystyle \sqrt{3}\). We would get \(\displaystyle \pi \sqrt{3}\) 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. 

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

The solution of \(\displaystyle \sqrt{4}\) is \(\displaystyle 2\), which can be written as \(\displaystyle \frac{2}1{}\). 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.