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 .

Since all the possible answers are square roots, we can square both the limits of our problem and all the possible answers. This allows us to see which number is correct. After squaring everything, we notice that we need a number between 81 and 100. The only possible answer given is 90. Now we can square root everything back, giving us our final answer of .

In order to rationalize, we will need to multiply the top and bottom by the denominator in order to eliminate the square root in the denominator.

Distribute and simplify the numerator.  Multiplying unlike numbers inside square roots will not eliminate the square root!

The answer is:  

Simplify the radicals

We notice that we have a complex number in the denominator. To get rid of this we multiply the numerator and denominator by the complex conjugate of the denominator.

Distribute across the numerator and multiply the binomials in the denominator. You may use the FOIL method.

We know that  so we replace  with -1

Combine like terms

Reduce and put in standard form

  or  

simplify the radicals

Distribute the negative

Combine like terms

Irrational numbers are nonrepeating decimals-- they cannot be written as fractions.

has only real numbers because the square root of 4 is 2, a rational number.

Factor the number -96

-1  2  2  2  2  2  3

The -1 come out the radical as an i. We look for pairs in the factors.

-1 ( 2  2 )( 2  2 ) 2  3

We see two pairs of 2, both can come out from under the radical . 

Multiply