If we add the same number with the same exponent four times, we can see the result will be four times the value. Assuming we couldn't recognize this at first glance, we can factor the common term our of our expression:

We cannot just simply multiply our bases together, however, because one of them has an exponent greater than 1. What we must do, then, is express 4 in a way that makes our two bases equal, which will allow us to add their exponents to arrive at our final answer:

As with any binomial with an exponent of 2, we can FOIL this expression to find its most simple form. As the expression is squared, we could rewrite it in the following way:

Now to FOIL the expression, we multiply the First terms, add the product of the Outside terms, then the product of the Inside terms, and finally the product of the Last terms, hence the acronym FOIL:

Here we need to recall a few simple properties of exponents. 

I)When we are dividing exponents of like base, we subtract the exponents.

II)When we are multiplying exponents of like base, we add the exponents.

III)When we are raising exponents to higher powers, we multiply the exponents.

I will apply and explain each of these properties as I solve this problem.

We begin with this:

I will begin by applying property II to the numerator of our fraction. Our like base is x, and the exponents we are adding are 3 and 4, so we get this:

Next we will apply property I to our fraction. Our like base is still x, but our exponents we are subtracting are 7 and 2, so we get:

Finally we apply property III to get our answer:

In order to simplify , first simplify each term:

Therefore, 

If  is not prime, it is factorable as 

where  and .

Therefore, we are looking for a whole number that is not the sum of two factors of 60. The integers that are such a sum are

Of the choices, only 18 is not a sum of factors of 60. It is the correct choice.

To simplify the expression start by simplifying each term.

From here, combine like terms.

To simplify this expression first simplify each expression.

From here combine like terms.

This is a simple problem that we can solve by rewriting the denominator as a product of its prime numbers. Since it is an odd number, we can start with 3.

3213 has three powers of 3. Let's try to divide the rest, 119, by 7.

119 is the product of 17 and 7, and since both of these factors are prime numbers, we are done calculating the factors of . Now, we can start canceling factors shared by the numerator and the denominator. After canceling shared factors, we are left with , which is the final answer.

Firstly, we must find the value of  for which  is true. You can solve this simply by testing out powers of :

4096 is 2 raised to the power of 12. Now we can easily say that . We can plug  in for  in  and solve:

 is a multiple of 7, so at the very least it includes a 7. Since  and  are both prime numbers, 7 is either  or . To make sure we have 49, the square of 7, into our product, we must take both the squares of  and  or , which is the final answer.