The reciprocal of  is the quotient of 1 and the number, or

This is not in scientific notation, so adjust:

You can simplify Column A first. When you're dividing with exponents and bases are the same, subtract the exponents. Therefore, it simplifies to x. We know that x is positive since it is greater than 1. X is greater than . Try plugging in a number to test. 25 is greater than , which is 5. Even 1.1 is greater than . Therefore, Column A is greater.

When you are adding and subtracting terms with exponents, you combine like terms. Since both columns have expressions with the same exponent throughout, you are good to just look at the coefficients. Remember, a coefficient is the number in front of a variable. Therefore, Column A is since . Column B is since . We can see that Column B is greater.

The perfect squares between 50 and 100 inclusive are

Their sum is 

The sum of the first ten perfect square integers:

The sum of the first five perfect cube integers:

(A) is greater.

The grayed portion of the Venn diagram corresponds to those integers which are not in any of , , or . Therefore, we eliminate any choices that are in any of the three sets.

 is the set of integers which end in 1 or 6; we can eliminate 166 and 176 immediately.

 is the set of perfect square integers; we can eliminate 144, since .

 is the set of integers which, when divided by 4, yields remainder 2. Since  , we can eliminate 154.

All four choices have been eliminated.

 is the union of sets  and  - that is, the set of all elements of  that are elements of either  or . We want all of the letters that fall in either circle, which from the diagram can be seen to be all of the letters except . Therefore, 

Based on the information given, you can construct the following Venn Diagram:

In order to find the overlap, you need to find out how many are in the circles together. This is easy. Subtract: . Now, since the overlap represents a duplication, you need to subtract out one of those duplicate values. Let's call that ; therefore, we know that:

Solving for , you get:

Based on the information given, you can draw the following Venn Diagram:

To solve this, remember that the total number of values in the two circles is:

(We must do this because of the overlap.  You need to subtract out one instance of that overlap.)

If we assign the value  for the unknown region, we know:

Based on the information, you can draw the following Venn Diagram:

It is very easy to solve for the number of plants that have green leaves but not large ones. This is merely . We find this by eliminating the large-leaved plants from the green ones (by subtracting the overlap from the green ones).